Often, my group picks (or acquires) a name for PC groups, mostly because it gives us something to call the campaign besides "Bob's Other Greyhawk Campaign."
For fantasy characters, a group name can be useful; a reputation sticks to a name easier, and reputations can be leveraged. And for superheroes, a group name is nigh-mandatory! (My group is currently trying to pick a name for a Mutants & Masterminds campaign.)
We've had:
- Riders of the Axe: they got that name when someone asked "Who are you people?", and the drunken dwarven priest (of an axe-wielding war god) that was trying to stay in his saddle said, "We're the Riders of the Axe, can't you see our horses and axes?"
- Company of the Sparrow: named after the copper coin of Keoland, after the second time a (randomly rolled) treasure was found to consist of thousands of copper coins. The characters almost all refuse to take any copper coins they find.
Aside from circumstance & arbitrary PC decision that happens to stick, other campaigns (mostly not D&D) have gotten names from the organization they work for (Shadowguard), the powers of the main PCs (darkness & psionics gave "Shadow Magic"), the unit's radio callsign (Coyote, Silver Dagger), or simple player decision ("We want to be the Centurions").
The other regular GM wants us to pick a name for his D&D campaign, but we haven't done much of note, besides get ran out of Gradsul by the thieves' guild, so it's been tough...