Nothing comparable to a Jersey diner in that area, but if you'll accept hole-in-the-wall, then I believe Lindy's Red Lion (on I at 21st) is open for breakfast. If it isn't, I know that Lindy's Bon Appetit downstairs from the Lion does breakfast; it's a carryout, though, but if the weather is nice, there are tons of places in that area to hang out and nosh.
(Your fallback option if nothing else is open would be to head to the food court at the GWU student center (21st between H & I). Or you can head up to Dupont Circle, where there are a lot more options. Pain Quotidien is good as Gigi007 says, but maybe a little too chichi for what you're looking for; depending on how early you need to eat, Luna Grill might be a good match for your diner needs.)
I can't think of any breakfast places in Foggy Bottom (too bad you don't like brunch 'cause there are a few good ones there!). If you don't mind walking a little or catching a bus (roughly 8 blocks away), in Georgetown, Furin's at 28th & M is diner-ish (it's a small family owned restaurant) and offers pretty standard breakfast fare (I'm not sure of their hours on Mondays, but they're definitely open for breakfast on weekends.). Le Pain Quotidien, a few doors down, might work too. It's not diner-ish, but it has great bread, waffles, cereal, etc and (I think) is open 7:00 am weekdays and 8:00 a.m. weekends. Both places have outdoor seating too.
I just remembered that Bread & Chocolate at 23rd and M serves a very good breakfast. It's not diner-like, but does offer a lot of good and reasonably priced breakfast dishes (and delicious blueberry muffins that I'm addicted to!).. It also has free WiFi and some outside tables.