El Morro, which is in old San Juan.
El Yunque (rainforest), which should be reasonably easy driving or cab distance from SJ.
If you like beaches, schlep out to Luquillo, not far east of SJ.
If you feel like renting a car and taking a day trip, I would do a loop that allowed me to see Hacienda Buena Vista, Ponce (including the art museum and downtown), and perhaps the tropical dry forest at Guanica (although the last might be too much in one day).
By way of low-hanging fruit, in old SJ not far from the cruise ship docks is a Don Q rum tasting room. If you are into high end rum, they sell their excellent Gran Anejo which is only available in PR and is truly world class.
"Neither my companion or I carry firearms on our persons. We depend on the goodwill of our fellow man and the forbearance of reptiles."
- English Bob