Rent a Jeep and explore the island.

Head to Shipwreck beach and take a picnic lunch and walk the beach exploring.

Drive through Garden of the Gods and go to the beach at the far western end of the island. There's a nice BIG beach there but unfortunately when we were there (late afternoon in late May) it was really REALLY windy and we all ended up getting sandblasted.

Take the Jeep on the Munro Trail. Awesome views at the top of the mountain (as long as you go early in the morning). It's quite a drive (dirt road, very bumpy and bouncy, quite fun).

Check out the petroglyphs.

Go into Lanai City and walk around the town square. It's very "old Hawaii" if you will and the people are fantastic.

Have dinner at the Hotel Lanai.

Play croquet on the lawn at The Lodge (and check out the orchid greenhouse too).

One suggestion, wear clothes you just don't give a crap about if you drive around on the dirt roads (of which most of the island has) because by the time you are back to the hotel, your clothes will be red with dirt as will your entire body. I had to SCRUB my body I am not kidding 3 times before I got the dirt off my face, arms and legs. I looked like a terra cotta statue