With this weather, head to St Augustine (about 30 minutes south) and America's oldest city. It's worth it. If you get there really early, head down south a couple of hours to Kennedy Space Center. Not much in Jax itself, except the beach and it's too cold for that. The zoo is good, though.
Oh yeah, just about 40 west is the Olustee battlefield where the South won one of its last battles.