This question actually reminded me that I had made some (minor) addenda/errata notes on my own thesis but somehow forgot to ever post them. I should get to that!
Anyway, you definitely should correct the error; it's bad for mathematics if serious errors propagate. It's a little embarrassing, yes, but it's not at all uncommon. It will be more embarrassing if someone else finds the error first, and extremely embarrassing (and maybe starting to be career-damaging) if you have to say "Oh yeah, I knew about that, but never corrected it because I was embarrassed."
Before doing anything, I would suggest you spend a few days seeing if you can fix the error. Maybe you can find a way to work around the gap in the proof. Maybe you can adjust the hypotheses so that you still have a theorem, though a weaker one. (Or maybe you can find a counterexample.) Figure out what other parts of the thesis are affected by this error, and if they need fixing as well.
It can get confusing if there are multiple versions of your thesis floating around, so rather than trying to revise it, I would write up a short erratum note, explaining whatever you have learned about the error and its fix (if any). You might as well also list any other typos or errata that you have found, references to relevant work that's been published (or discovered by you) since your thesis was written, and anything else you would like to add.
Post the addendum on arXiv. Then, wherever your thesis is posted (your web site, the alternative preprint server you mention, the university's official site if possible), post a note saying "Addendum posted at arXiv:1234.5678". If you can't add notes or comments on the site itself, but can upload a revised version of the thesis, just add a page at the beginning with a reference to the addendum.
(And I'll echo JeffE's suggestion: it would be nice to post your thesis on arXiv itself as well. The "Comments" field would be a good place to reference the addendum.)