Can I submit my old PhD thesis to arxiv with the date set to its original submission date? I have completed my PhD 10 years ago, and back then I was not sure if it was ok with my home institution to have it in arxiv. I then realized it's fine and it would actually be good as my institution never cared to set up an open database of their own PhD theses. My question is: is there a way to post it on arxiv with the proper date, so that it shows on a search but it does not show among the most recent ones?

EDIT: given the popularity of the question, would it make sense to allow the ability to backdate ONLY PhD theses? It seems like my experience is a shared one, and I can't imagine anybody in her/his own mind lying on when she/he got her/his PhD.
I don't think so.

The "proper date" for an eprint is the date you upload it to the arxiv.  If the arxiv allowed people to back date their work (by up to ten years!), then it would at best confuse and at worst totally ruin the use of the arxiv to sort out chronology and claims of priority.  So while I don't know that they don't allow this, I have never seen an example, and I hope that they don't.

What you can do, of course, is to include in the comments that this is your thesis from 10 years ago.  If you do that then everything seems clear.  If you are worried that your thesis appearing on the feed/list of new papers will cause it to get more attention....wait, isn't that why you are posting it to the arxiv in the first place?