For not-too-expensive stuff for the kids, any ABC store will present a multitude of choices. They have things like key chains and little necklaces made of shells that don't cost too much.
You can ride a city bus to Ala Moana shopping center, about 15 minutes from the main Honolulu hotels in central Waikiki by bus. You'll find the usual mall rat stores, as well as a larger and slightly cheaper Food Pantry and a gigantic food court. Wander through Shirokawa, a Hawaiian department store that sells all things Japanese and much other Asian stuff. Look around the dry teas section, or the household goods, for ceramics and daily use things for the kitchen from Japan.
To get there by bus, on Kuhio Avenue, Saratoga Road or Kalia Road heading away from Diamond Head, board either 8 Ala Moana Center, 19 Airport-Hickam, 20 Airport- Pearlridge, 42 Ewa Beach or 58 Waikiki-Ala Moana.
To return, board 8, 19 or 20 Waikiki Beach and Hotels on Kona Street. Buses run approximately 10 minutes apart.
Travel time is approximately 15-20 minutes.
I have never found anything worth doing, with or without kids, at the Royal Hawaiian Shopping Center, right in Waikiki. The stores are aimed at what used to be hordes of free spending Japanese tourists. Well, the economy in Japan went to heck and the hordes are fewer in number and the place feels lonely. An expensive kinda lonely, but lonely nonetheless. However, if you suddenly jolt awake in Hawaii and need a Rolex NOW!, this is the place to buy it.