I buy eye of round quite often. I use it in stews (Norwegian Sailor Stew comes to mind, as does the venison-ale stew I make--I think I've posted both recipes and Whiskadoodle tweaked the Norwegian Sailor Stew recipe)--both of these stews contain beer/ale. You could marinate the meat overnight in a marinade containing beer, wine, or brandy. You could also marinate the meat in black tea to tenderize it.
Check out this site:
This is one of my favorites (the brandy tenderizes the meat):
Brandied Beef - Recipe - Cooks.com
With homemade beef stock it is fantastic. I think my recipe is in one of the Bon Appetit cookbooks I have, but this sounds identical, except the title was Braised Beef with Orange or something like that. I have used round steak with the fat removed, can't think why eye of round wouldn't work.