I'd have to say her name and some random facts. About what, I don't know. Maybe architecture facts like you suggested or maybe something about her features, blonde hair, tan skin. I wouldn't be surprised if at the Roman Camp, Percy saw an owl or something and related it to Annabeth's eyes.

I would love for Percy to spit out random architecture facts in the middle of the quest!

I think that there might be that; then her face, but that's pretty much it, I believe. (although the factoids is just me dreaming up cute Percabeth scenes, so technically my answer is 'her face')

I was almost tempted to pick the Random Factios choice...

I think that he knows her name, face, age, relationship, and random facts that keep coming to him throughout the course of the story.

ie.
"I will destroy you!" Clytius yelled.
Percy blurted out, "The Hoover Dam is over 200 meters tall, built in the 1930s, and holds 5 million cubic acres of water!"
Clytius paused. "What?"
"Uh... I don't know."
"This is no time to be spouting architectural facts!" Hazel hissed.

Haha, if it was actually like that in the book, I'm gonna die of laughter...

Oh gods, I hear you!

And yeah; it will come back, but as of right now. Like, as he runs from Stheno and Euryale what does he know?

Probably her name, face, age, and relationship. Or if not relationship, then at least Percy knows that Annabeth has a special connection to him...

it said that percy remered she was inportant to him.

It's like that warm feeling where your reminded of something special. It's like when he hears her name it's like he's opening up a book of all their memories that he can't quite see or hear or replay in his head, but he can feel the special energy.

^That makes me feel warm and fuzzy... And it's probably going to be true. Very true.

I think that everytime he sees a child of Minerva he'll be frustrated to death because it'll taunt him so much. Just being in California because they had such strong moments there will frustrate him. Seeing a big and nice building will leave him muttering to himself either architecture facts or questions.

But I think he remembers her face, for example he may remember her 'curls like princess'or her 'stormy grey eyes'. And all the looks she's ever given him.

I think he's gonna know she holds answers to his past, like
Thalia did for Jason. He knows he needs to find her. I think it's gonna be random factors for two reasons:
1. Rick Riordan's writing style, which is humor humor and a little more humor. As mentioned by partypony higher up, it will add a lot of cause for humor.
2. Percy's most likely to remember the most insignificant details because he can't see the big picture very well (see Percabeth predicament).

so yeah...

True that, true that... I'm thinking of changing my answer... But I do think he'll remember her face and a bunch of random faacts.

He'll recognize her on sight, he knows he loves her, he knows she's around his age, he might know she has family issues, he knows he'd give the world for her, maybe that she calls him Seaweed Brain, some random architecture facts(like in TTC, he see's a monument and blurts out a factoid Annabeth told him) but doesn't know how he knows


Thats it I think