So, let's run through the rest of the team, yes?
Gonna ignore Red Arrow and Zatanna for now...neither has spent enough time on the team for us to see if they have a Huge Secret. Plus, Zatanna has a big enough angst source without needing a secret, and Red Arrow doesn't seem to care about friendship.
Robin: Nothing obvious, but he's a Bat-clan member. I think we can take horrible trust-destroying secrets as a given here.
Aqualad: In love with his best friend's girlfriend. Also, not so much a secret he's hiding as something that doesn't come up much, his gills mark him as Impure and subject him to racist persecution in Atlantis. Being half-human doesn't help him there either. As of #14 of the comic, Superboy and Miss Martian know his secret, but he doesn't yet know they know.
Miss Martian: Really a White Martian. Psymon knows her secret and taunts her about it.
Artemis: Her entire family is supervillains, although Mom served her time and has retired. Obviously, her father and sister know her secret and taunt her about it.
Superboy: He's half-Luthor, is taking gifts from Luthor and keeping this all secret from the others. Luthor knows, of course, and taunts him about it. His tendency to have metaphorical sticky fingers around the property of the powerful (Wolf, Supercycle) doesn't help.