If you're taking along your riding gear, you'll probably be about the only rider there wearing any protective clothing. Hardly anyone wears a helmet there.
+1 on that.
I grew up on Oahu, and never even considered road riding until I moved to California. It was pretty rare to see anyone in anything more than a helmet and maybe a mesh jacket.
One consideration is it will be hot, humid, and there will be a good deal of traffic on the freeways, and with development, all of this is getting worse all the time. Lots of squiddy riding going on and the cagers there can be pretty lax in their attention, awareness, and execution. Another thing is there's a lot of Fast and Furious blowback there, so be careful of those douchebags in modded Civics and Supras etc.
That said, make sure you get the hell out of Honolulu; take the H-1 (if I remember correctly) and head east for Diamond Head, Sandy Beach, and Koko Head, or even Kaena Point. If you head west, there's the North Shore, Turtle Bay, and you can loop the island that way. It'll take about two hours and it's a really nice drive once you get out of traffic. If you're staying in town, Tantalus is nice, as it's already been mentioned, but again, watch out for the racers up there. It's seriously an idiot's training ground.
Another consideration is to head west from downtown and cross the island on the H-3 headed over to the Windward side. It's a nice stretch and that side of the island is somewhat less touristy.
Again, it's gonna be hot and humid. Oh, and because of the winter rains and heavy traffic combined with a shit budget for road maintenance, there are a good deal of potholes, loose spots, and tar-and-pebble areas, so keep an eye out.
It's definitely a beautiful place. Hope you enjoy yourself there, and ride safe.
Oh, and sorry about being no help with the rental question.
--Roy
