Oscan-speaking peoples, but I don't know how exactly.
Ok, brainstorming time.
We know that Oscan was still spoken in probably multiethnic 79 AC Pompeii, which implies that the language was preserved by the mountaineers for at least another century or two though we have no Oscan inscriptions from the late empire, but the similarity with Latin makes its extinction inevitable at some point or another - as long as continued Roman rule remains.
Which means that, maybe, it is Roman rule that we have to wipe out first. Let the socii win the Social War somehow, have an Italic, Oscan-speaking state right on Rome's doorstep in southern Italy for some time. Maybe the socii can sign an alliance with Mithridates VI in time (this was actually planned OTL but iirc the envoys reached him only by the time the war ended in 88 BC though unrest would continue until 82 when the former socii sided with Marius and almost destroyed Rome at the Colline Gate). Maybe the Romans refuse to hand out citizenship until too late, and IOTL it was citizenship which divided the allied front since some were fighting to acquire it and some others simply wanted out of their alliance with Rome.
How Rome solves its inevitable manpower issues considering that southern Italians made up half of its forces is up for debate. Another issue that presents itself is the connection with Greek cities such as Taras and Rhegion, which never revolted.
Maybe they make up for it with Cisalpine Gauls, Illyrians and perhaps Spanish tribes - after dealing with Sertorious, or maybe they extend citizenship to non-Romans and Italics earlier, learning from the mistake of the Social War. Either way, Rome makes up for this loss and, in due time - a century and a half or less - it reacquires control over southern Italy, first through a protectorate a la Egypt, Judea, Pergamon, Bosporus you name it, and then by annexing it.
This is precious time for the Oscan language: aside from its administrative use, a proper literature has the chance to emerge (the logical starting point is an account of the Social War) giving it the needed literary prestige, maybe the Italic alphabet is replaced with the Greek one, and by the time Rome has reacquired control over the south, there is a strong enough local identity.
Romanization does happen either through adoption of Latin by some elites or by immigration of Roman citizens, but it is a limited and possibly reversible kind, similar to Albania's or Greece's where its Romance population is limited to Vlachs.
This brings us to the post-Roman period, eventually. Any power intent on ruling all of the peninsula will use Latin as its official language, just as in Gaul and Spain for example, but a smaller state might preserve its legitimacy by using Oscan as a working language. I can picture southern Italy under Odoacer, Theodoric, the early Byzantines and the unified Lombard kingdom using Latin, but the moment the Lombard kingdom fractures, the dukes of Benevento (see: Malies) first and then Salerno (see: Irna), Capua (see: Kapu) etc might consider the use of the local language. Even the Normans might follow suit.
Another outcome of this is a different perception of what constitutes Italy. To us it is logical to assume the entire peninsula is Italy, but in fact the name made its way north only over time, and the existence of a state called Italia right on Rome's doorstep inevitably alters the perception of what constitutes Italy. By now, we would call only the south Italy, and maybe the whole peninsula including it with another name such as Hesperia.
I cooked this up in like 20 minutes as I typed.