Well, there's the Contemporary Museum:
The Contemporary Museum Going there has the added bonus that you can do the drive up Tantalus / Round Top and see some of the best views the city has to offer.
The Honolulu Academy of Arts has many galleries. One permanent gallery has contemporary art. They also have rotating exhibits... a recent one was on photography. Not sure what's there now.
http://www.honoluluacademy.org/cmsha...emy/index.aspx They also have a nice cafe, so you can eat lunch while you're there. If you're here the last Friday of the month, you can check out Art After Dark at the Academy. Food, drink, live music, and open galleries. Lots of fun.
The Hawaii State Art Museum
Hawaii State Art Museum -- Hawaii Museums Association Database -- has a mix of galleries... I've only been once for a fundraiser and didn't get to all of the galleries, but there seemed to be some modern stuff there. Another great option for lunch is the cafe there. Plus, they're near downtown / Chinatown which has lots of pre- and post- museum exploring potential. If you're here the first Friday of the month, you can go t "First Friday". The State Art Museum is open, and there's music out on the lawn. Plus, the Chinatown art galleries stay open late and often host special events.
UH has art exhibits, often of student and faculty work, at the art building. It can be hit and miss, but gives you an excuse to go see the UH campus if you want to do that.
Also, the Tennent Art Foundation (on the way, depending on which way you go, to the Contemporary) is the former house of Madge Tennent and houses a lot of her work.
Tennent Art Foundation Gallery -- Hawaii Museums Association Database --