I'd say Yakitori Yakyudori in Hillcrest, but they're only open Wed-Sunday now. The location in Kearny Mesa is open tonight, but only til midnight and is too far, if you don't have a car (or feel like driving 20-30 minutes).
There really aren't so many options for late night in San Diego. Of those...there are the following, though none of them would be considered on any "best restaurant" list outside of their own categories (burger, pizza, mexican food). But they're all sit-down, or a bar.
The Waterfront is in Little Italy right near the airport. Great bar burger/menu. Historic(ish) bar for San Diego, so it's kind of divey. Some local taps/bottles as well.
There's also Santana's fast food Mexican right next door to the Waterfront, which is 24 hours, and a particularly good location for Santana's. Nothing beats some good Mexican fastfood.
Basic Urban kitchen is open late in the Gaslamp. Great pizza and salads, but that's it on the menu.
La Puerta in the Gaslamp. Open til 1am. Excellent tequilas and great street-style Mexican food in a sit-down restaurant/bar. If I had to pick one from this list it would be to eat here. Mondays are half-off drinks and appetizers all night as well. So, that's a nice bonus.
Starlite is great too, but kind of in an obscurish part of town for a visitor. Great interior, and you have to get a Mule to drink.
3rd Corner in OB. Not only best, but the only option unless you count 24 hour taco shops.
Edited to add: seems to be closed Mondays, of course.
Might want to try Starlight - near airport as well. Website claims "Late Night Dining" but that means different things to different people.