At Darling Harbour in Sydney there is The Sydney Wildlife Centre which gives you a glimpse of some of the native animals in Australia, and next door is the Sydney Aquarium. As an older person I enjoyed both even though I have had many trips to Aus and been to many wildlife parks. Taronga Zoo is also interesting.
There are several beaches that are ideal for children - Manly and Bondi are just 2 I can think of - and easily accessed from the city via a bus or ferries.
For other ideas google Sydney for children - I found quite a few sites and ideas listed there.
I can't comment on Fiji and Hawaii as I have never been to either.

In Sydney, I recall there being a nice 'ride' at the top of the Centrepoint tower.
Not sure if its still there but my kids thought it was great.

Its been a while now but, we had a great visit to Fox studios, again, don't know if they still do the tours?

If these are lost in time, apologies, I really need to go back.

Muck

In March you could still visit some beaches and the weather will still be sunny but a lot milder and perhaps not so 'burny'; I second Taronga Zoo and a ferry ride across the harbour to get there. Manly with the Oceanarium, the two beaches and the Corso (mid-priced shopping area linking the two beaches); also a ferry ride to get there. The Australian Museum (near Hyde Park) has a lot of good exhibitions, including some really good ones on prehistoric animals and Australian animals that are now extinct. Very kid-friendly.

Agree about Sydney Tower (whatever they are calling it nowadays - it's possibly called AMP Tower, they keep changing the name. One caveat - it's fantastic but expensive. Even more expensive is the Bridge Climb, which I have heard is fabulous (my husband has done it, and so has my friend with her 9 and 10 yr old sons). You can get some of the same effect by walking the bridge on the traffic level as there is a footpath, which is free, and you can actually go inside one of the pylons, which is a museum.

Also if you are with a 9 yr old girl, if she likes shopping, you might have a look at the shops off the Pitt St Mall. In the Glasshouse (a shopping centre) is a shop with cool clothes for 8+ girls, Urban Angel (it's the sister shop of Pumpkin Patch, which is for much younger children, and has just lovely clothes). I am not wowed by the new Westfield (a new shopping centre), as I'm not into high-end designers, but Myer and David Jones are standard department stores you might visit.

Lavandula

Re Bridge climb, I recall there was a lower age limit.
My Daughter wasn't allowed up in 2001 when she was 11. I think she just missed it by a few months as she had to be 12 years old.
Had to do it next time in 2005, and she loved it.

But I just checked and the min age has changed to 10 yrs old.

Muck

I returned from Australia to Toronto in March. This was my second trip. No matter how you arrange it, it is a VERY long flight. The last time we broke up the trip by staying one day in LA on the way and several days in Hawaii on the way back.

Based on that experience I feel that one day is not sufficient to deal with the jet lag and LA is too close to home to stop. You still have approximately a 15 hour flight ahead of you and many time changes. Fiji might be a suitable alternative but make sure you stay long enough to let your body clock catch up.

If it is a case of just one night stopover, I would bite the bullet and continue on to your destination. To me, anything less than three nights isn't enough to deal with the jet lag.

I celebrated my 75th birthday by doing the Sydney Harbour Bridge Climb on March 10, 2012 - an experience of a lifetime. I don't think they allow someone as young as your grandchild to climb but it would be worth enquiring about yourself if you can arrange it. Although it is "pricey" it was well worth it, especially when compared to the costs of other excursions and activities. The Bridge Climb is 3 1/2 hours in total. I recommend it highly - even for great-grandmothers like me!

I can echo prinret's comments about jet lag. I've been to Australia 4 times, and each time I seem to take a little less time to get over jet lag. But It still takes at least four days before I feel like myself, and that long or longer before I can sleep through the night. If possible I would suggest you save the busy days in Sydney till the end of the trip, or consider some chill out time enroute at the island of your choice.

Good to hear of your bridge climb, Prinret. Have something planned for your 80th?