They've pegged him at 1178 BC. I was displeased with this number initially, because Troy 6 is dated at 1700-1250 B.C., and those 72 years would be far too long for Odysseus to fight, get lost, and come home to Penelope outside of an old age home. ("Alas! I should have stayed Calypso!") But having just pulled my trusty Troy book by guide Mustafa Askin off the shelves, I realized that Mustafa's theory (as well as some other scholars) is that Troy 7a is actually Homer's Troy, not Troy 6. That city dates 1250 to 1180. That falls pretty neatly into line with the eclipse of Odysseus.
As a note, while researching this, I came up with five offspring of Odysseus that were conceived during his travels: three with Circe (who turned his men into pigs) and two with Calypso. Those Greek heroes did like to get around--but, acknowledging that I'm applying modern standards on mythic Greece--I feel like Penelope kinda got the raw end of that deal.
* I specify here because the printed version of The Odyssey we all have access to is most certainly literature--but it without a doubt stems from earlier oral mythic traditions. So it's somewhere in that nebulous continuum