In episode 11 season 2 of One Punch Man, Garou's right eye turns red and also his hair does. Is there an official name for this form?
I probably don't think so he had been given a name to his new personality. I have read the manga and i don't recall any new name given to him
This is just part of how Garou gets progressively more and more jacked up as the arc progresses, but yet he keeps on trucking and getting stronger. It's really hard not to understate the sheer level of worked-over this guy achieves. The following happens to him in chapter 68 of the webcomic, for example, and this is after already having suffered numerous injuries in preceding battles:
The red you refer to is just blood. A blood vessel has popped in his eye and he's bled so much that it's begun to matte and stain his hair. More clearly, you'll notice that immediately before the reveal of his red hair and eye that he still has white hair, and then rapidly wipes his hands in his hair. He's had a lot of blood on his hands and arms, bleeding notably out of them at least since deflecting Death Gatling's attack; we also see he's still rupturing blood all along his arms when he squares up against Bang.
The animation is slightly misleading after this, as you still see a far shot of him with white hair after this before the close-up reveal, nor does it show any color changes while he's doing it. I think this was all done for the purpose of the dramatic reveal being a dramatic reveal. There's no other reason for wiping his hands on his hair other than to transfer blood to his hair, either strictly for a makeshift cleaning of his hands or, more likely, because he knows it serves to make him seem more monstrous.
In the webcomic, by chapter 82 he ends up a full-on horror freak show from the battle damage alone.
He does later on achieve what appear to be legitimate monster-like transformations, but at this point in the story, if the manga/anime aren't going to seriously alter the nature of his progression, then all you are seeing is the accumulation of (severe) battle damage.