I'm with nanatravel on the Hotel Milano if you're looking for good value. I've never stayed there myself but just sort of tripped over it and began recommending it to people. So far, no complaints and good comments.
I am particularly fond of the El Convento, which is pricier but includes a large continental breakfast and wine and cheese every evening. It's a nice place to meet nice people.
Old San Juan is a very safe city these days (wasn't always)
You might want to cruise over to the archived feature articles. There's one called Bienvenidos a San Juan that should give you and your daughter a lot of options.
A cab from the airport to downtown was $18 the last time we were there which wasn't all that long ago.
Ten un buen viaje !!! (Have a good trip, that's an order !!!)
pamda -- CM First-Time Cruisers Editor and sometimes Puerto Rican