According to Trinitarians, what is the Biblical basis that some doctrine is a mystery?

God is incomprehensible, because He is infinitely beyond us. He is a Trinity, so it follows that His Trinitarian nature would be incomprehensible to men. Scripture has many verses which support this notion that mankind will never fully understand the Divine Essence and all of the truths concerning it.

“Can you fathom the mysteries of God? Can you probe the limits of the Almighty? They are higher than the heavens above—what can you do? They are deeper than the depths below—what can you know? Their measure is longer than the earth and wider than the sea. Job 11:7-9

He hath made every thing beautiful in his time: also he hath set the world in their heart, so that no man can find out the work that God maketh from the beginning to the end. Ecclesiastes 3:11

Who has measured the waters in the hollow of his hand, or with the breadth of his hand marked off the heavens? Who has held the dust of the earth in a basket, or weighed the mountains on the scales and the hills in a balance? Who can fathom the Spirit[d] of the Lord, or instruct the Lord as his counselor? Whom did the Lord consult to enlighten him, and who taught him the right way? Who was it that taught him knowledge, or showed him the path of understanding? Surely the nations are like a drop in a bucket; they are regarded as dust on the scales; he weighs the islands as though they were fine dust. Lebanon is not sufficient for altar fires, nor its animals enough for burnt offerings. Before him all the nations are as nothing; they are regarded by him as worthless and less than nothing. With whom, then, will you compare God? To what image will you liken him? Isaiah 40:12-18

Trent Horn offers a good explanation of the Catholic understanding of the Trinity as mystery for further reading.

First, God does this by intention:

It is the glory of God to conceal things, but the glory of kings is to search things out. (Proverbs 25:2)

Second, prophecy tells us that some mysteries will be revealed. If there were no mysteries concealed, then there would be none to reveal.

1 Then I saw in the right hand of him who was seated on the throne a scroll written within and on the back, sealed with seven seals. 2 And I saw a mighty angel proclaiming with a loud voice, “Who is worthy to open the scroll and break its seals?” 3 And no one in heaven or on earth or under the earth was able to open the scroll or to look into it, 4 and I began to weep loudly because no one was found worthy to open the scroll or to look into it. 5 And one of the elders said to me, “Weep no more; behold, the Lion of the tribe of Judah, the Root of David, has conquered, so that he can open the scroll and its seven seals.” (Revelation 5:1-5)

By being analogized as a scroll, the information currently hidden that Jesus, the worthy Lamb, shall unseal, must be intelligible. Once unsealed, the church will be able to read it. This likely means that there are mysteries hidden in the Bible that the Holy Spirit will teach us, once those seals are broken. The seals will not be broken all at once, so this means a steady stream of mysteries will be clarified for us over some indeterminate period of time.

Third, just as there is a mystery to God's ways, there is a mystery to Satan's:

4 The woman was arrayed in purple and scarlet, and adorned with gold and jewels and pearls, holding in her hand a golden cup full of abominations and the impurities of her sexual immorality. 5 And on her forehead was written a name of mystery: “Babylon the great, mother of prostitutes and of earth's abominations.” (Revelation 17:4-5)

Fourth, the progressive nature of God's revelation is emphasized by the several references in Revelation to the seven spirits of God going out into the earth. Isaiah names those seven Spirits:

And the Spirit of the Lord shall rest upon him, the Spirit of wisdom and understanding, the Spirit of counsel and might, the Spirit of knowledge and the fear of the Lord. (Isaiah 11:2)

At least four of those spirits communicate what could be considered mysterious information, namely the spirits of wisdom, understanding, counsel and knowledge.

Since the question addresses the Trinity, even that doctrine was a mystery, disclosed to the church after the time of the Apostles.

According to Trinitarians, what is the Biblical basis that some doctrine is a mystery?

The term mystery simply signifies that which is unknowable, or valuable knowledge that is kept secret. Mystery comes from the Greek mysterion, from myein, "to shut", "to close".

This term signifies in general that which is unknowable, or valuable knowledge that is kept secret.

Notion of mystery in Scripture and in theology

The Old-Testament versions use the word mysterion as an equivalent for the Hebrew sôd "secret" (Proverbs 20:19; Judith 2:2; Sirach 22:27; 2 Maccabees 13:21). In the New Testament the word mystery is applied ordinarily to the sublime revelation of the Gospel (Matthew 13:11; Colossians 2:2; 1 Timothy 3:9; 1 Corinthians 15:51), and to the Incarnation and life of the Saviour and His manifestation by the preaching of the Apostles (Romans 16:25; Ephesians 3:4; 6:19; Colossians 1:26; 4:3).

In conformity with the usage of the inspired writers of the New Testament, theologians give the name mystery to revealed truths that surpass the powers of natural reason. Mystery, therefore, in its strict theological sense is not synonymous with the incomprehensible, since all that we know is incomprehensible, i.e., not adequately comprehensible as to its inner being; nor with the unknowable, since many things merely natural are accidentally unknowable, on account of their inaccessibility, e.g., things that are future, remote, or hidden. In its strict sense a mystery is a supernatural truth, one that of its very nature lies above the finite intelligence.

Theologians distinguish two classes of supernatural mysteries: the absolute (or theological) and the relative. An absolute mystery is a truth whose existence or possibility could not be discovered by a creature, and whose essence (inner substantial being) can be expressed by the finite mind only in terms of analogy, e.g., the Trinity. A relative mystery is a truth whose innermost nature alone (e.g., many of the Divine attributes), or whose existence alone (e.g., the positive ceremonial precepts of the Old Law), exceeds the natural knowing power of the creature.

Catholic doctrine

The existence of theological mysteries is a doctrine of Catholic faith defined by the Vatican Council, which declares: "If any one say that in Divine Revelation there are contained no mysteries properly so called (vera et proprie dicta mysteria), but that through reason rightly developed (per rationem rite excultam) all the dogmas of faith can be understood and demonstrated from natural principles: let him be anathema" (Sess. III, Canons, 4. De fide et Ratione, 1). This teaching is clearly explained in Scripture. The principal proof text, which was cited in part by the Vatican Council, is 1 Corinthians 2. Shorter passages are especially Ephesians 3:4-9; Colossians 1:26-27; Matthew 11:25-27; John 1:17-18. These texts speak of a mystery of God, which only infinite wisdom can understand, namely, the designs of Divine Providence and the inner life of the Godhead (see also Wisdom 9:16-17; Romans 11:33-36). Tradition abounds with testimonies that support this teaching. In the Brief "Gravissimas Inter" (Denzinger, "Enchiridion", ed. Bannwart, nn. 1666-74), Pius IX defends the doctrine of supernatural mystery by many citations from the works of the Fathers. Numerous other patristic texts that bear on the same question are quoted and explained in Kleutgen's "Die Theologie der Vorzeit", II, 75 sq.; V, 220 sq.; and in Schäzler's "Neue Untersuchungen über das Dogma von der Gnade" (Mainz, 1867), 466 sq. The manifold excellence of Christian revelation offers many theological arguments for the existence of supernatural mysteries (cf. Scheeben, "Dogmatik", I, 24).

Mystery (Catholic Encyclopaedia)

Further biblical basis can be seen in the following:

• Great is the Lord, and greatly to be praised, and his greatness is unsearchable. - Ps. 145:3

• Behold, these are but the outskirts of his ways, and how small a whisper do we hear of him! But the thunder of his power who can understand? - Job 26:14

• For my thoughts are not your thoughts, neither are your ways my ways, declares the Lord. For as the heavens are higher than the earth, so are my ways higher than your ways and my thoughts than your thoughts. - Isaiah 55:8–9

• Oh, the depth of the riches and wisdom and knowledge of God! How unsearchable are his judgments and how inscrutable his ways! "For who has known the mind of the Lord, or who has been his counselor?" - Rom. 11:33–34. (Also cf. Job 42:1–6; Ps. 139:6, 17–18; 147:5; Isa. 57:15; 1 Cor. 2:10–11; 1 Tim. 6:13–16)