I have felt unhappy for a long time with the name 'Old Testament' to describe the first section of the Christian Bible. I use it but simply for lack of an alternative.
It is not that I don't believe in the idea that there is an old and a new covenant/testament (the Bible is clear on that); my niggle is simply with using those terms to describe the two parts of Scripture. Those books do not constitute the old and new covenants respectively so at best it is slightly confusing.
It also tends to close down the wide-ranging significance of the two sections of the Bible by squashing everything in them into the distinction between old and new covenants. But if we read them with such a narrow lense we would miss so much!
Perhaps we might say that they bear witness to the old and the new covenants and there is a lot to be said for that (though it is Jeremiah - not a NT writer - who introduces the idea of a new covenant) but we still have the problem of an overly narrow descriptive name.
Now if used with care the language of OT and NT is OK but in real life it is hard to escape the feeling that for many Christians OT = old, not-very-relevant, etc. So there is an inclination to avoid it. Or an inclination to dismiss it with a simple, "but that's in the Old Testament." I wonder if a different name would mitigate such inclinations.
NT authors describes Israel's holy texts as 'scripture' or 'the scriptures' (not the Old Testament). It was the creation of the NT canon that required renaming Israel's scriptures so as to differentiate them from the new ones.
But the problem of renaming raises the thorny problem of finding a better alternative.
Some people refer to the OT as 'the Hebrew Bible'. I have some problems with that.
1. being picky, not all of it is in Hebrew.
2. more importantly, for a Christian there is only one Bible and the OT is a part of it. To call it 'the Hebrew Bible' seems to require that the NT be a second Bible (the Greek Bible? [but then, might parts of it have originally been in Hebrew or Aramaic?]).
3. also 'the Hebrew Bible' tends to refer to the OT with the books organized in the order found in medieval and modern Jewish Bibles. Christian Bibles follow a different (though still Jewish) order. So the OT is not 'the Hebrew Bible' as such if we think organization is of any relevance.
Some people refer to the OT as 'the Jewish Bible'. There are two problems here.
1. See 3 above.
2. This implies that the NT is composed of Christian (and not Jewish) texts. But that is pure anachronism. Most NT authors were Jews and most of them were writing for Jewish followers of Jesus (Paul, being the most notable exception). NT texts are Jewish texts (and even Luke-Acts, which may well be by a Gentile, is very Jewish in its theology).
A similar problem accompanies 'Israel's scriptures'. Setting aside debates on the import of the word 'Israel' in this name (which parts of Israel recognized which scriptures and when?) we also have the problem that it suggests that NT texts are not 'for Israel'. But such a view may well be anachronistic reading back later distinctions between Jews and Christian alien in the first century. I guess we could refer to the OT as Jewish scriptures accepted by all Jews and the NT as 'extra scriptures accepted by some Jews'. Apart from being cumbersome it has the effect of undermining the value of NT texts (questionable ones) in relation to OT (accepted by all).
So what then?
Tanak, the Jewish name for the books (minus apocrypha) is a big improvement. It simply designates the three sections - Torah, Prophets, and Writings. It has the downside that it presupposes an organization of the canon that is not identical with that found in the Christian OT. Strictly speaking the OT is not the Tanak (even if the books within both are identical - if we set aside discussions on the apocrypha).
Perhaps Christians could draw inspiration from the Jewish approach and make up a word like
The PHoPP (Pentateuch, History, Poetry, Prophets)
The THaPP (Torah, History, Pentateuch, Prophets)
The GALoR (Gospels, Acts, Letters, Revelation), or GAER (if you prefer 'epistles' to 'letters'.
To be honest ... I can't see that working!!!
John Goldingay suggests 'first testament' and 'second testament'. That's a little better but I'm still niggled by the 'testament' part.
I prefer the NT word 'the scriptures'. Obviously, once the NT canon is formed, we need a way to extend that term to cover the NT but also to find a way to distinguish it from the OT. How about 'first scriptures' and 'second scriptures' or 'earlier scriptures' and 'later scriptures'.
On the other hand, such chronological terminology feels a little theologically neutered. The advantage of Old Testamanet and New Testament is that it is a highly theological way of describing the collections.