Does the league name change/mean anything?

They serve no purpose, it's simply there because it's similar to the old system (ELO system), where every player was on the same list (and you could be ranked at 12 231 532th)

It has no influence on matchmaking (but it does appear to try to put you in the same league as your friends [e.g. if your friend is in Silver III league xyz, and you're promoted to Silver it's more likely that you'll be put in league xyz])

Other than that it has no influence at all. Regarding the names it's always a champion name randomly paired with a noun, several of which are listed in the old forums. Champion names are not directly associated with a noun.