It's been two decades since we were in the toddler business, but you should start your research with the 101ThingsToDo.com website. When you're at the airport's baggage claim area, pick up a copy of "This Week Oahu" for an events calendar (and a good Waikiki map). There are plenty of other free maps & guides in the racks, and plenty in them for a toddler to look at too. You can download the HulaCopter and Happy Hour Pal apps on a smartphone if you're looking for discounts on Waikiki events. (Happy Hour Pal does restaurants as well as bars & clubs.) There's also an activity calendar in the Midweek newspaper (Midweek.com).
I'd skip straight to sunscreen in the SPF 30-70 ranges. Waterproof & sweatproof, none of that spray variety-- you want the type that's applied with a trowel. Floppy hats. You'll also want reef socks for everyone so that you don't cut your feet on the odd rock or lump of old coral. A three-year-old could do fine with a t-shirt in the water, but a rashguard will help guard against sunburn and be a lot easier to rinse/dry.
Everything in Waikiki is expensive, including groceries. You'll want a bus or taxi to the Iwilei Costco (on Alakawa) or the Wal-Mart or Sam's Club on Ke'eamoku. Souvenirs are cheapest by taking the 30-45-minute bus ride out to Aloha Stadium on Wednesday/Saturday/Sunday mornings for the Aloha Stadium Swap Meet. Emergency diaper resupply can be found at an ABC store or a 7-11 or the Longs Drugstore in Ala Moana Shopping Center. You can buy floaty innertubes and beach gear (and rashguards) here, but if you have room in your luggage then it's a lot cheaper to pack it.
The beach park between the Hale Koa Hotel and the Hilton Hawaiian Village has plenty of shade and a small playground. Between that and trips across the beach to the ocean, you could set up for the entire day. If you have a military or DoD civilian ID then you can get into the Hale Koa's toddler-friendly kiddie pool, but the beach across the sidewalk is just as good. Otherwise it's easier to roam the grounds of the Hilton. Moving down further into Waikiki, you could enjoy the beach bustle around Duke Kahanamoku's statue. You could also do a stroller walk around the grounds of the Hilton (critters around the pond & pools) and the Fort DeRussy Park on the other side of the Hale Koa. A late afternoon stroll along the Ala Wai Canal has plenty of canoe paddlers to watch, but I wouldn't splash around in the Ala Wai itself-- lots of runoff from the mountains is drained through there.
With a bus or taxi ride you could get around to the island's east side. I don't think a three-year-old is ready to tackle the Diamond Head hike unless you're willing to carry him. Makapu'u Point is an easier walk, more stroller-friendly, and you can bail just as quickly. You could spend a day at Hanauma Bay (closed on Tuesdays) but a toddler won't appreciate the difference between that and Waikiki.
While you're at the Swap Meet you could also go across the street to the USS ARIZONA Memorial. All of the Memorial is free, including the visitor center's museum and the surrounding grounds. The USS MISSOURI Memorial and the Pacific Aviation Museum both charge admission but are are more kid-friendly than the USS BOWFIN Submarine Museum. The BOWFIN itself, though, will get rave reviews. While you're waiting on the boat for the ARIZONA, you can wander around the Submarine Museum grounds.
Other cheap toddler entertainment: the Honolulu Zoo is the best zoo for 2500 miles in any direction. Same for the Waikiki Aquarium. Bishop Museum has a lot of kid-friendly displays and activities, and you'll want to ask about their special-events schedule with shows and crafts demonstrations. I know you can buy an annual pass for Bishop Museum, and you can probably do something similar for the Zoo & Aquarium.
There were Monday-night aerial fireworks on Waikiki Beach this week. I'm not sure whether that's every Monday or a special event. The Hilton also has a Friday-night fireworks display. Depending on your toddler, this is either an invitation or a warning.
I think taking a three-year-old to a luau is a setup for a meltdown. It depends on their temperament, how late they can stay up, and how they feel about flashy lights & loud noises. (Ours wasn't ready until she was six years old.) The Wailana Coffee House (corner of Ala Moana & Kalia) is a fairly kid-friendly restaurant open 24/7. ClifP will kid me about its kitschy decor but it's cheap and the food is yummy.
