Ps 8:2 reads as follows:
Masoretic Text (Hebrew): From the mouths of infants and nursing babies You have established strength Because of Your enemies, To do away with the enemy and the revengeful. The highlighted word is עֹז (oz), meaning (BDB) noun masculine strength, might; — absolute ׳ע Judges 9:51 +, עוֺז Psalm 84:6 +, עָזֿ Isaiah 26:1, once עָ֑ז Genesis 49:3; construct עֹז Micah 5:3 +, עָזֿ Psalm 28:8; suffix עָזִּי Exodus 15:2 2t., עֻזִּי Psalm 28:7 +; עָזְּךָ Psalm 21:2 2t., עֻזְּךָ Psalm 66:3 +; suffix 1 plural עוּזֵנוּ Psalm 81:2; 3 masculine plural עֻזָּמוּ Psalm 89:18, etc.; — strength, might (usually in poetry, 44 t. Psalms)
Septuagint (= LXX, Greek): Out of the mouth of babes and sucklings hast thou perfected praise, because of thine enemies; that thou mightest put down the enemy and avenger. The highlighted word is αἶνος (ainos) which means (BDAG) "praise"; Thayer - "praise, laudatory discourse".
In Matt 21:16, the Greek text of the NT uses exactly the same word as the LXX. So, Jesus quoted the LXX exactly.
Therefore, we should ask, Why does the LXX differ from the Masoretic (Hebrew) Text? We do not know the answer to this but it is very likely one of the following reasons:
- The exemplar for the LXX differed here (and many other places) from the Masoretic text
- The LXX translators took liberties with the translation. (This is much less likely but instances of this can be shown in other places.)
Therefore, the most likely reason that the LXX differed from the Masoretic text is that the Hebrew text the LXX used differed as in mast places where the two diverge.
There may also be a third possibility as noted by Ellicott and his comments on Ps 8:2 -
Ordained strength . . .—At the first glance, the LXX. translation, as quoted in Matthew 21:16 (see Note, New Testament Commentary), “Thou
hast perfected praise,” seems to be correct, from a comparison with
Psalm 29:1, where strength translates the same Hebrew word, and
plainly means homage. This expresses, doubtless, part of the thought
of the poet, that in a child’s simple and innocent wonder lies the
truest worship; that God accomplishes the greatest things and reveals
His glory by means of the weakest instruments—a thought which was
seized upon by our Lord to condemn the want of spirituality in the
scribes and Pharisees. But the context, speaking the language of war,
seems to demand the primitive meaning, stronghold or defence. The
truth which the Bible proclaims of the innate divinity of man, his
essential likeness to God, is the principal subject of the poet; and
in the princely heart of innocence of an unspoilt child he sees, as
Wordsworth saw, its confirmation. “Trailing clouds of glory do we
come, From God who is our home.” Such a proof is strong even against
the noisy clamour of apostate men, who rebel against the Divine
government, and lay upon God the blame of their aberration from His
order. “His merry babbling mouth provides a defence of the Creator
against all the calumnies of the foe” (Ewald). Others think rather of
the faculty of speech, and the wonder and glory of it.
Benson and Barnes offer a similar explanation.