As far as I remember, there isn't any name for it. It is an unnamed island.
The LotR Wikia says the same:
The sea also contains an unnamed island that is about thirty miles along the southeastern and northwestern coasts and twenty upon the northeastern and southwestern coasts.
Also, it must have been wooded. (thainsbook).
There has been a similar discussion on TheTolkienForum. A user contacted The Tolkien Society with the question, and here is the answer:
Hello. Sorry about the delay. A staw poll revealed that in a very
early version of the map, the Sea of Rhun had a large island in the
middle. (Unfeasibly large, given the scale of the Sea, which is very
big.) By the time the map hit publication, this had vanished and the
Those who have looked carefully at the various maps point out that
there are similar dotty effects at the Mouths of Sirion and near Balar
(Silmarillion), and references to "sands" in the Sil. The dots in the
Sea of Rhun (on scale) could quite easily represent a constellation of
small islands, but are more likely to mean sands, swamp or sand bars.
(Actually small islands would make better geography on that scale.)
There appears to be no actual reference to what is going on there.
There is a discussion of this also at TheOneRing.net which adds some
details worth having. I don't however understand Quickbeam's reference
to CRT "finding later manuscripts etc." as CRT drew up this map in
co-operation with JRR while JRR was still alive and well and writing
the books - there have AFAIK been no revisions by CRT to these maps on
the basis of stuff he found after his father's death, when he started
editing the unpublished mss. Indeed, it would be against his whole
attitude of respect for his father's decisions to start "editing" LotR
material subsequent to publication. I have asked QB what he means, but
have not had a reply. As this was some time ago and you are waiting,
this is my conclusion!
I would say "be careful of taking any craft with deep draft through
these waters. You may get stuck."
Which, TBH, doesn't add much more than that there is no name to the island.
The discussion Helen references may be worth quoting for posterity:
I’ve been wondering about this for years, and now have found a few
bits of evidence in the History of Middle-earth Volume VII that gives
us a better clue.
Most all of the maps we see in today’s editions of LOTR come from the
hand of Christopher Tolkien; redrafted from his father’s originals.
Professor Tolkien altered and added to his maps as his work on the
story progressed. His original working map (the "First Map") was a
conglomeration of sections and glued bits of paper that he fussed with
for many years: "It represents an evolution, rather than a fixed state
of the geography." There was a later version called the "1943 map"
that Christopher created.
Comparing these two maps, Christopher points out a well-forested
island his father drew in the middle of the Sea of Rhûnaer, which
would later be called the Sea of Rhûn. "The island in the Sea is
coloured green on the First Map, and on the 1943 map is marked as
wooded." That’s your island right there, but in later maps it seems to
have vanished, replaced with an array of dots.
I don’t know exactly why Christopher changed it, but I suspect that
later manuscripts and maps he found showed the removal of the island.
Perhaps the discrepancy caused him to leave the dots to indicate what
was a "questionable area."
This seems to be the clearest answer, if still vague.