As far as I'm aware, all our names for the ancient tribes of Britain, Gaul, and Germania come from Greeks and Romans like Ptolemy and Tacitus, so we end up with names like Picti, Iceni, and Boii. But do we know what the Celtic tribes called themselves?
A small anecdote, but an important one: Julius Caesar in his Gallic Wars mentions in the first few lines that the Romans called the folks who lived in Gallia the Gauls (Galli) but they called themselves Celts (Celtae).
In later portions from his Commentarii De Bello Gallico, he'll mention that several tribes chant their own names as they run into battle, which he records (albeit in a Latin transliteration).
Plutarch also mentions the custom of chanting one's own tribe name when rushing into battle, in this case, the Ambrones. (See section 19 for the relevant passage).
Thus, we do have some of the original names, as transliterated in Latin/Greek.
Sources:
Julius Caesar, Commentarii De Bello Gallico.
Plutarch, Life of Marius
Edit: added some links and a bit more information
Often after the Roman conquest the pre-Roman divisions were maintained as socio-political units. So, for example, the area of the pre-Roman Dobunni because the civitas Dubunnorum. This is used in personal as well as public settings, so we see inscriptions in which people identify themselves as of the civitas Dubonnorum. The names are Latinized, but certainly reflective of the pre-Roman naming.