We've had various party names come and go over the years, some better than others:
The Enterprise Company (yes, named after the flippin' starship) - royal decree forced adventuring groups to incorporate (in effect) and this is the best they could come up with
Company of the Green Cloak - so named because during a session someone noticed that by sheer chance every mini in the party had been painted with a green cloak
The Lightbringers - this might have been a good idea if the party hadn't then immediately scattered to the four winds and formed new parties
Company of the Broken Soliloquy - a party started calling themselves this one day and I-as-DM still have no idea why, as I missed whatever conversation led up to it
Most of the time, though, when our games have multiple parties running concurrently we just use their usual geographical location* within the game world to refer to them e.g. the Northern party, the Eastern party, etc.; or if they're also being played concurrently on different nights it'll just be "the Friday group", "the Sunday group" etc.
* - which is right at the moment a tad confusing because both the game I play in and the game I DM have two main parties, in each case very clearly a "Northern" and a "Southern".
The Exiled - there were a number of places that for some reason wouldn't have us back
That just tells me you're doing it right!
Lan-"having been in parties that had no towns left in the realm that would allow us in, I can sympathize"-efan