Saturday and Sunday will both be crowded, so check the weather and use that as your guide for when to go. The island is beautiful in both the rain and sun, but if you want to see the bluest blues and the greenest greens, check the weather report. And, the mornings are clearer than afternoons, so take an AM tour rather than a PM tour. Like all tropical areas, the heat and humidity builds up during the day so if it is going to rain, it usually does in the afternoon.
The Polynesian Cultural Center is run by the Mormon Church, is usually closed on Sundays. I do think they have the least commercial luau...but no alcohol on their premises if that is important to you!
Hanauma is popular any day of the week (closed Tues), but since a lot of the tourists go home on Sunday, Mondays are definitely less crowded, but see BOX JELLYFISH SCHEDULE BELOW. Go absolutely first thing in the morning, as you have the best chance to see sea turtles before too many people scare them back out to sea and I have usually seen the moray eels earlier than later (and also get a parking spot!). Also, people kick up the sand so the water starts clouding up towards the middle of the day. The other thing to check for is box jellyfish, which are very poisonous. This is a relatively new phenomenon (when I was young we didn't have these) no one knows why. In any case, they come close to shore to breed 7-11 days after a full moon, and are expected to peak in May 17-19th. So, if you are going this weekend, go snorkeling before Sunday the 17th. There are now torch (evening) tours of the reef, you may want to consider seeing the lagoon at night.
http://www.808jellyfish.com/808jELLYFISH2009/2009%20CALENDARS/Mayarrivalcalendar2009.htm