That's...actually a really good idea that I've never even though of before. Having the option to name your group if you please while a simple touch would only add that much more to immersion and role playing.
Having the option of being the "Hearth Alliance" a group of civilized pacifist Orlans dedicated to bringing knowledge and acceptance of their kind to the world would be much more engaging than being "a group of 6 people" (just a random on the spot example)
Yeah, that's exactly what I was thinking. It would add more depth to the lore while adding more role-playing options.