If I was forming a band (and the good news is, I'm not about to) there's just one place I would go looking for name. Brewer's Dictionary of Phrase and Fable. It is a goldmine. Here's some random possibilities, and the band one might form (if one was forming a band).
Inexpressibles: art rock, obviously.
Marine Store: industrial strength indie
Mari Lwyd: ambient-ethnic-folk
Maul of Monks: goth-core
Rochelle Salt: riot grrrl revisited
Ruffian's Hall: Hmmm, not sure about this one. Maybe they aren't, either
Sea Lawyer: smart-arse pop (think TISM meets Loudon Wainwright III)
Sugar Daddy: r 'n' b, apparently
Dr Syntax, pub rock with art rock pretensions
Synecdoche: instrumental, post-prog
The Seven Against Thebes: Liverpudlian trip-hop
Tokyo Rose: I actually saw them in Perth in the 19080s. They were terrible.
Wayzgoose: John Butler wannabe
Praying-wheel: indie wimp-pop
See, Brewer's Dictionary of Phrase and Fable. For all your musical needs.