What is the significance of Koizumi noticing Kyon's mole at the end of Lone Island Syndrome?
I don't think there was this ever an explanation made in the anime... was there any context given in the light novel?
I don't have any hard evidence to back it up, but I always assumed that it was at that point that Koizumi actually started taking Kyon seriously. Instead of just looking at what he seems to be at first glance, he started peering into him at greater depth. And, since it's mostly a comedy series, that greater depth involves a mole.
In the light Novel at least:
there is a scenario where the cast encounters well, a fake cast (fake Kyon, fake Haruhi, etc.) Kyon manages to deduce that fake Mikuru was fake because there was no mole on her breast. Most likely we are shown the mole of Kyon to also understand the fact on how Koizumi will be able to distinguish the fake Kyon from the real Kyon in this incident (because Koizumi encounters the fake Kyon).
Of course, note that this is just speculation too, as the Light Novel has a first person perspective (so we don't actually read about Koizumi seeing the mole, so therefore the above also won't be read about). However, it's quite likely that KyoAnim added it because of this case.
The scene can be found in:
The Rampage of Haruhi Suzumiya, The Snow Mountain Syndrome chapter. To quote:
A deep breath comes forth from the depths of my throat.
I jump out of my bed as if I had a spring in me.
Look closely! Why is it that I didn't notice this? The person before me is not 'my Asahina-san'. I should have known better than anyone else. Didn't I do this last time to verify for authenticity? You would know after looking at 'that spot' on Asahina-san.
"Who are you?"
-There is no mole on the left breast of this Asahina-san.
He [Koizumi] stares right at my [Kyon's] face.
[Koizumi]: "You appeared in my room as well. The appearance might have been you, but the behaviour was just terrifying... anyways, you did things that you wouldn't do."