In the New Testament the word "mystery" often implies a "secret rite" in the Greek (see Thayer's lexicon also here). In the Latin Vulgate, it is often translated as sacramentum from which the English word "sacrament" is derived. LDS people mean "sacrament" when we say "ordinance." The two words are really synonymous. In LDS terminology, referring to the Lord's Supper as "the Sacrament" isn't technically complete. It is "the sacrament of the Lord's supper" or "the ordinance of the Lord's supper." Baptism, confirmation, priesthood ordination, etc are all "sacraments" or "ordinances." To avoid the confusion of referring to the Lord's Super as "the sacrament," the Church translation of the scriptures into Spanish refer to la santa cena (the Holy Dinner) instead of to el sacramento (the sacrament).
In many older texts and often considered the more authentic reading, 1 Cor. 2:1 says that "I, brethren, when I came to you, came not with excellency of speech or of wisdom, declaring unto you the mystery of God." Later texts changed it to "Testimony." Mystery fits better in the context. Let me show you why.
For example, verse 6: "Howbeit we speak wisdom among them that are perfect (teleoi, that is, mature, or the initiated), yet not the wisdom of this world, not of the princes of the world, that come to naught. But we speak the wisdom of God in a mystery, even hidden wisdom, which God ordained before the world unto our glory..."
Then in chapter 3 he repeats this theme, saying "I, brethren, [when I was among you] could not speak unto you as unto spiritual, but as unto carnal, even as unto babes in Christ. I have fed you with milk, and not with meat (adult food): for hitherto ye were not able to bear it (that is, until now you couldn't bear it), neither yet now are ye able (and you still can't!)" (1 Cor. 3:1-2).
In the next chapter, Paul writes, "Let a man so account of us, as of the ministers of Christ, and the stewards of the mysteries of God" (1 Cor. 4:1).
So, It appears that Paul had in his mind two classes of Saints. The teleoi (or mature, the initiated, or "perfect") who had given to them hidden wisdom in a mystery (sacrament, ordinance, secret ritual) who could receive the adult food. The second class was those who hadn't received the hidden wisdom in a mystery because they were not yet ready. The people in Corinthians who had been members of the Church already for years fell into the second class, according to Paul's epistle to them. That's pretty fascinating.
I think this topic is also hinted at by the author of Hebrews (whom I believe to be Paul).
For when by reason of the time ye ought to be teachers, ye have need again that some one teach you the rudiments of the first principles of the oracles of God; and are become such as have need of milk, and not of solid food. For every one that partaketh of milk is without experience of the word of righteousness; for he is a babe. But solid food is for fullgrown men [teleoi], even those who by reason of use have their senses exercised to discern good and evil (Revised Version. Hebrews 5:12-14).
Another fascinating reference to this is in Ephesians 5. In this context Paul speaks of marriage and the Church:
Wives, submit yourselves unto your own husbands, as unto the Lord. For the husband is the head of the wife, even as Christ is the head of the church: and he is the saviour of the body. Therefore as the church is subject unto Christ, so let the wives be to their own husbands in every thing. Husbands, love your wives, even as Christ also loved the church, and gave himself for it; That he might sanctify and cleanse it with the washing of water by the word, That he might present it to himself a glorious church, not having spot, or wrinkle, or any such thing; but that it should be holy and without blemish. So ought men to love their wives as their own bodies. He that loveth his wife loveth himself. For no man ever yet hated his own flesh; but nourisheth and cherisheth it, even as the Lord the church: For we are members of his body, of his flesh, and of his bones. For this cause shall a man leave his father and mother, and shall be joined unto his wife, and they two shall be one flesh.
This is a great mystery (Ephesians 5:22-32).
Paul refers to marriage as a "great sacrament" and also speaks of the church as a bride without "wrinkle" or "spot" painting an image of special ceremonial wedding clothing.
Anyway, this continues after the New Testament. Here's the info I've gathered so far on the topic. Please read the quotes carefully before you start saying they refer to baptism or the Eucharist (some may, indeed, but not all for reasons that will be apparent to the careful reader). I think there is some real clear indicators that the "mysteries" were often other ordinances now long lost (except to the Latter-day Saints).
For a mystery ought to be most faithfully concealed and covered, especially by us, who bear the name of faith. (Lactantius Ante-Nicene Fathers 7:221)
And Peter said: “We remember that our Lord and Teacher, commanding us, said, ‘Keep the mysteries for me and the sons of my house.’ Wherefore also He explained to His disciples privately the mysteries of the kingdom of heaven. But to you who do battle with us, and examine into nothing else but out statements, whether they be true or false, it would be impious to state the hidden truths.” (Peter, Clementine Homilies, Ante-Nicene Fathers 8:336)
For the most sublime truths are best honoured by means of silence. (Peter, Clementine Recognitions, Ante-Nicene Fathers. 8:83)
But if [Simon Magus] remains wrapped up and polluted in those sins which are manifestly such, it does not become me to speak to him at all of the more secret and sacred things of divine knowledge (gnosis), but rather to protest and confront him, that he cease from sin, and cleanse his actions from vice. But if he insinuate himself, and lead us on to speak what he, while he acts improperly, ought not to hear, it will be our part to parry him cautiously. For not to answer him at all does not seem proper, for the sake of the hearers, lest haply they may think that we decline the contest through want of ability to answer him, and so their faith may be injured through their misunderstanding of our purpose. (Peter, Clementine Recognitions, Ante-Nicene Fathers. 8:98)
But the same writer [Clement of Alexandria] in the seventh book of the same work, relates also the following things concerning him: “The Lord after his resurrection imparted knowledge (gnosis) to James the Just and to John and Peter, and they imparted it to the rest of the apostles, and the rest of the apostles to the seventy, of whom Barnabas was one...” (Eusebius. The Church History of Eusebius 2.1.4. Nicene and Post-Nicene Fathers, Series Two. 1:104)
The science of nature, then, or rather observation, as contained in the gnostic tradition according to the rule of truth, depends on the discussion concerning cosmogony, ascending thence to the department of theology. (Clement of Alexandria. Nicene and Post-Nicene Fathers, Series 2. 8:42)
Wherefore also all men are His; some through knowledge (gnosis), and others not yet so; and some as friends, some as faithful servants, some as servants merely. This is the Teacher, who trains the Gnostic by mysteries, and the believer by good hopes, and the hard of heart by corrective discipline through sensible operation. (Clement of Alexandria, Ante-Nicene Fathers. 2:524)
[T]he mysteries are not exhibited incontinently to all and sundry, but only after certain purifications and previous instructions. (Clement of Alexandria, Ante-Nicene Fathers. 2:449)
Now, in answer to such statements, we say that it is not the same thing to invite those who are sick in soul to be cured, and those who are in health to the knowledge and study of divine things. We, however, keeping both these things in view, at first invite all men to be healed, and exhort those who are sinners to come to the consideration of the doctrines which teach men not to sin…And when those who have been turned towards virtue have made progress, and have shown that they have been purified by the word, and have led as far as they can a better life, then and not before do we invite them to participation in our mysteries. “For we speak wisdom among them that are perfect.”…[W]hoever is pure not only from all defilement, but from what are regarded as lesser transgressions, let him be boldly initiated in the mysteries of Jesus, which properly are made known only to the holy and the pure…He who acts as initiator, according to the precepts of Jesus, will say to those who have been purified in heart, “He whose soul has, for a long time, been conscious of no evil, and especially since he yielded himself to the healing of the word, let such an one hear the doctrines which were spoken in private by Jesus to His genuine disciples.”… [Celsus] does not know the difference between inviting the wicked to be healed, and initiating those already purified into the sacred mysteries! Not to participation in mysteries, then, and to fellowship in the wisdom hidden in a mystery, which God ordained before the world to the glory of His saints, do we invite the wicked man, and the thief, and the housebreaker, and the poisoner, and the committer of sacrilege, and the plunderer of the dead, and all those others whom Celsus may enumerate in his exaggerated style, but such as these we invite to be healed…God the Word was sent, indeed, as a physician to sinners, but as a teacher of divine mysteries to those who are already pure and who sin no more. (Origen, Ante-Nicene Fathers. 4:487-489)
In these circumstances, to speak of the Christian doctrine as a secret system, is altogether absurd. But that there should be certain doctrines, not made known to the multitude, which are (revealed) after the exoteric ones have been taught, is not a peculiarity of Christianity alone, but also of philosophic systems, in which certain truths are exoteric and others esoteric. (Origen. Ante-Nicene Fathers 4:399)
Nicene and Post-Nicene Fathers:
Of the beliefs and practices whether generally accepted or publicly enjoined which are preserved in the Church we possess derived from written teaching; others we have received delivered to us “in a mystery” by the tradition of the Apostles... (Basil of Caesarea, Nicene and Post-Nicene Fathers, Series 2. 8:40-41)
In the same manner the Apostles and Fathers who laid down laws for the Church from the beginning thus guarded the awful dignity of the mysteries in secrecy and silence, for what is bruited abroad random among the common folk is no mystery at all. This is the reason for our tradition of unwritten precepts and practices, that the knowledge of our dogmas may not become neglected and contemned by the multitude through familiarity. “Dogma” and “Kerugma” are two distinct things; the former is observed in silence; the latter is proclaimed to all the world. (Basil of Caesarea, Nicene and Post-Nicene Fathers, Series 2. 8:42)
We ought not then to parade the holy mysteries before the uninitiated, lest the heathen in their ignorance deride them, and the Catechumens being over-curious be offended. (Athanasius, Nicene and Post-Nicene Fathers, Series 2. 4:106)
But first I wish to remind you who are initiated of the response, which on that evening they who introduce you to the mysteries bid you make; and then I will also explain the saying of Paul: so this likewise will be clearer to you; we after all the other things adding this which Paul now saith [in 1 Corinthians 15:29]. And I desire indeed expressly to utter it, but I dare not on account of the uninitiated; for these add a difficulty to our exposition, compelling us either not to speak clearly or to declare unto them the ineffable mysteries. Nevertheless, as I may be able, I will speak as through a veil. (John Chrysostom. Nicene and Post-Nicene Fathers, Series One. 12:244)
What may all of this mean for mainstream Christians today who often teach that all an individual needs to know to be saved is that Jesus is his or her Savior? Where are the gnosis and musterion today?