In Daggerfall it is spelled Sumurset Isles and almost everywhere else it's spelled Summerset Isles. So which is correct?

I'd say the latest rendition that is Summerset Ilse

Soo-moor-set is how I say it, so I spell it Sumurset. The UESP lists three official spellings, I think. Pick the one you like.

In Oblivion the NPCs pronounce it "Summerset Isles", like the season.

I think it was Summurset in Arena, Daggerfall, and Sumurset in Redguard.

I like Summurset :3

But Summerset is the most up-to-date version, speaking in order of games. But, of course, in TES 5, we'll be seeing sammarset.

Use whatever one you want.

Somersault! The Somersault Isles! It turns out the Altmer were acrobats and not mages all along.

Given that Summerset seems to be the standard more recently, I'd go with that, Sumurset is most likely either an outdated, or at least now rarely used spelling within the universe, or it's been retconned into Summerset.