In my experience your only option with 4x4 is a Jeep Wrangler. Many companies rent them, but they are usually the most expensive thing you can get. Some of the rental places will tell you not to engage the four wheel drive (they don't want you going 'off-road'), but it is not actually disabled. (You are thus better off not telling them you will being using the four wheel drive.) Do not get a non-Jeep SUV as these are generally not 4-wheel drive.
Cuatro-por-cuatro means four-by-four, but you will probably find that agents working at rental car place speak perfect English. (Many people on PR, particularly in rural areas, do not speak English well if at all - however people working in the tourist industries often speak excellently.)
Charlie Car Rental (closest location is Avenida Isla Verde, I believe) and Target Car Rental (closest location off PR-26 in Santurce) are home-grown chains that probably also rent Jeeps. They may or may not be cheaper than the U.S. companies. At least Charlie Car does airport pickup. Target Car Rental is a $20 taxi ride from the airport (although they may also pick you up).
You may not need four wheel drive. If you are being extremely adventurous on the high mountain roads the gradients can be very steep, in which case four wheel drive is reassuring, although front wheel in a standard car would probably cut it. Also, many of the rural roads are paved but in bad shape (potholes, etc.) so having higher clearance in an SUV or Jeep is helpful, although rarely absolutely necessary (you can also carefully navigate the potholes if the roads are really rough). Where on the island do you plan to go so I could give better feedback from my own experience?
