Unless you want to do a heli tour over the volcano in Hilio, or the coast and landscape of Kauai, a rental car allows you the freedom to move as fast or slow as you like... so that would be my recommendation. (Both/either heli tours are wonderful though).
We just took this cruise on the Island Princess in January of 2006, and loved it so much that we are booked to repeat it in March 2008! Different ship next time - Diamond Princess - same itinerary.
We did Princess excursions, as follows:
Hilo - Volcanoes National Park - definitely a great place to see, whether you take the ship's excursion or you choose to rent a car. If you go on your own, be sure to also visit the Thurston Lava Tube.
Kona - We did the glass bottom boat excursion, because neither of us snorkle, but we wanted to see some underwater stuff. It was nice and we saw a lot of corals, fish, and some sea turtles. After the tour we walked around Kona in the historic area near the pier. There's lots to see in that area.
Honolulu - Another definite "do" is the USS Arizona Memorial. The ship's excursion we took also included a city tour and a trip up into the hills to the Nu Pali Overlook; the views of the north side if the island are gorgeous. You could do these on your own, but traffic in Honolulu, especially on the freeways was a nightmare. If you routinely drive L.A. freeways or similar you might be OK. Our tour driver had been a Honolulu police officer and really knew his way around, and was used to the traffic.
Kauai - we did not take an excursion here. We chose to ride the free (tips only) shuttle buses into town to Hilo Hatties, and another to a beach near the pier, where there also was a cluster of nice shops. The beach was in front of a medium sized hotel, and was quiet. This was a very relaxing day.
Maui - We did the excursion for whale watching, and it was amazing. Seeing all those whales was a great experience. After the excursion we wandered around the shops on Front Street (I think that was the name of it) which is right near the pier. We found a little jewelry store where we each bought a 14 K "whale tail" charm for a very reasonable price. Made a great souvenir.
I sincerely believe you will have a wonderful time on the cruise. Even though you only have one day per island, it is a very nice introduction to Hawaii. If you happen to have a Hawaii historian/naturalist onboard, by the name of Bob Basso, try to attend each of his presentations. He is very informative and interesting. Each of his presentations on our cruise was well attended, most times filling up the Princess Theater. He did presentations almost every sea day; they only last an hour, so they don't take a big chunk out of your day.
2004 Coral Princess - Panama Canal
2005 Sun Princess - Eastern Caribbean
1/06 Island Princess - Hawaii
2/07 Sea Princess - Southern Caribbean
11/07 Golden Princess - Mexican Riviera
3/08 Diamond Princess - Hawaii
11/9/08 Tahitan Princess - Tahiti & French Polynesia
9/13/09 Star Princess - Alaska
3/14/10 Golden Princess - Hawaii
If you enjoy snorkeling, the Nal Pali boat ride is fantastic. The only gamble is the weather. You leave out of a beach-front dock about a half hour out of the port area in a zodiac type boat. Very large (for a raft), with an awning and latter for entering and leaving the ocean. This was a Princess tour and absolutely fantastic. Check out my photos in the gallery.
In Hilo I HIGHLY recommend going to Akaka Falls! It's breathtaking! My shore excursion was through NCL and we also went to Nani Mau Gardens, however I was not that impressed by Nani Mau Gardens but the trip was worth it to see Akaka Falls!
In Maui I recommend going to the Iao Needle.
We also did the Atlantis Submarine tour on one of the islands and were not impressed, the colors of the fish were very muted so we ended up paying more $ to go to an aquarium, which was much more fun!