We don't know anywhere near as much as we would like about Carthaginian names- our written sources only rarely mention anything about family names or groupings (so we know of, for example, the Magonids, and, a bit less certainly, the Barcids). Certainly the Carthaginians took some pride in their family and the achievements of their ancestors, but they have left so little behind that we can say almost nothing about that aspect of their culture with any certainty. Carthaginians tended to only use single names, and appear to have only drawn from a relatively small pool of these, so it can be quite hard to differentiate them- note, for example, the prolific amount of Hanno's and Hasdrubal's down Carthaginian history.

Hoyos mentions that we know of about 500 different Carthaginian names from various stele (inscribed stones), and that most are derived, in some way or another, from the name of the gods. Names that referenced Baal, Astarte and Melqart were comparatively common, though Tanit does not appear to be utilised in this fashion. We encounter Abdmilqart/Habdmilqart ("servant of Melqart"), Abdastart ("servant of Astarte"), Bodmilqart ("in Melqart's service"), Gersakun ("fear of Sakun", a fairly obscure deity), Saponibaal ("may Baal watch over me"), Zakarbaal ("Baal, remember me"). Many Carthaginian names, including the above, were altered into more familiar forms by the Greek and Roman writers who recorded them, so Abdmilqart/Habdmilqart became Hamilcar; Abdastart became Bostar; Bodmilqart became Bomilcar; Gersakun became Gisco/Gesco, Saponibaal became Sophoniba; Zakarbaal became Sicherbas/Acherbas.

On the other hand, some Punic names remained mostly the same: Hannibal ("Baal be gracious to me"), Hanno ("grace be to him"), Himilco ("Milkot/Melqart is my brother"), Marharbal ("hasten, Baal") and Mago (shortened form of Magonbaal, "may Baal grant") made the jump relatively unscathed by linguistic alteration.

Dexter Hoyos, "The Carthaginians" (Routledge, 2010)