Saturday breakfast around Logan Circle
A couple other ideas below, but I don't know hours.
I'm a regular at The Diner, but it's not a real diner, if you're a purist. For that, you could consider Florida Grill (truly greasy and been there forever), which should be open -- or Ben's Chili Bowl, for that matter -- if they're within your walkable range.
Cafe St Ex
1847 14th St NW, Washington, DC 20009
1633 P St NW, Washington, DC 20036
Florida Avenue Grill
1100 Florida Ave NW, Washington, DC 20009
If you are looking for a diner I would head up to Adams Morgan (not that far of a walk) to The Diner. Around 8am on a Saturday it's not too busy. They have the diner basics (great pancakes, waffles, eggs, omlets etc). It's open 24/7 and has a great, friendly staff (a little less officiant when it gets busy but at 8am they are usually fine).
Not sure how early Tabard Inn is open but that would be a great spot to enjoy a cold fall morning breakfast.
Hudson in Dupont/West End opens I think around 10am and they have a full breakfast/brunch.
You should also see if Beacon Bar and Grille and Dupont Grill are open at that hour. They probably are since they are hotel restaurants.