I didn't realize how few of the places we usually eat are BYOB; I guess we like alcohol provided with our meal. But a few selections:
Tanoshii (5547 N Clark) - a sushi place run by the inventive "Sushi Mike". The thing to do here is not to order off the menu, but just to tell your server your general likes and dislikes (and allergies, I suppose) and how much you want to spend and let the sushi chef go to town. They rarely disappoint.
Hama Matsu (5143 North Clark) - just a regular sushi place, but nice.
Icosium Kafe (5200 N Clark) - great Moroccan crepes, both savory and sweet. If you're trying to Google it, note that they seem to randomly interchange Cs and Ks (Ikosium Cafe, etc).
Pho Xe Tang - Tank Restaurant (4953 N Broadway) - big noisy Vietnamese.
Apart Pizza (5624 N Broadway) - This is a bit of a fudge. I suppose, technically, you could eat in and BYOB at Apart, but the atomosphere is nil. But I put it on the list because it's really good pizza and you could order in and drink whatever you want. I suppose under that criteria, Ras Dashen and Indie Cafe are both great fancy-order-in-and-drink-your-own (FOIADYO?) options.