We Do Not Know
I haven't found any information on the names of any of the other islands in the Stepstones but I have found out that George R. R. Martin once commented that he hadn't named one of the islands yet (although I haven't confirmed this it's likely this post is referring to the most south western island). If he hadn't named one of them it's likely he hadn't named more than that one too. Though the post is from 2006 it's likely he still hasn't named any of the others.
I never got to ask a question during the Q&A. But i sliped in a quick one during the book signings. I asked about the unmarked island near the Dorne. His answer was that it was just one of the stepstones and he hadn't even come up with an actual name for it. And it didn't have a lord as of yet. And apparently there are several more islands of simalar size that are extend further east. So now i'm not even sure it is part of westeros.
So Spake Martin, Canadian Signing Tour (Toronto)
Having tweeted Elio and Linda, the co-authors of The World of Ice and Fire, they have said they they don't know of any other names either.
Hello! I've seen that two of the islands in the Stepstones have been named, Bloodstone and Grey Gallows. Also that Tyrosh is sometimes considered a part of them but isn't really one. Do we know the names of any of the other Stepstones? Thanks!
No other names that I'm aware of.
It's worth noting that the Stepstones are fairly recent, geographically, as they are the remnants the Land Bridge that connected Westeros with Essos.
The Dornishmen boast that theirs is the oldest of the Seven Kingdoms of Westeros. This is true, after a fashion. Unlike the Andals, who came later, the First Men were not seafarers. They came to Westeros not on longships but afoot, over the land bridge from Essos—the remnants of which exist today only as the Stepstones and the Broken Arm of Dorne. Walking or riding, the eastern shores of Dorne would inevitably have been where they first set foot upon Westerosi soil.
The World of Ice and Fire, Dorne
On an interesting side note an island was added to the west of Bloodstone in The Lands of Ice and Fire that wasn't present in the Map of the Free Cities from A Dance with Dragons.