It turns out that Solomon Airlines has cancelled the Tuesday return flight, so I may get to stay in charming Honiaria

instead, or take the newly back in the air after bankruptcy Air Nauru.
Did they cancel that specific return flight or all tuesday return flights? I just came back from the Solomons 3 days ago (was there for a month, 2nd time this year), and the thursday return flight is a horrid schedule as it leaves at 7am. This means you leave the ship at 4:30 am, and you cant dive at all the day before. This was the first time in many trips there that we had that return flight. I hope thats not permanent 
Are you going to Sydney for a reason? It may be better to fly either USA->Fiji->Solomons or USA->Brisbane->Solomons. Saves you 2 flights between Brisbane and Sydney. Brisbane has enough sights to keep you busy for a few days, and the weather will be reasonable enough so you dont have to take many extra clothes. I can suggest at least going to Lone Pine animal park for a day. I would definately stay at least 2 days in Australia if you can, because you wouldnt be the first one to have their luggage come late. Give yourself some time to fix any problems that arise.
If you absolutely have to stay in Honiara, stay at the Mendana Hotel and get a good book. But i'd avoid it.
If you're keen on seeing Sydney, then by all means go see Sydney. It's a huge city with plenty of places to visit. I usually stay around Circular Quay which gives you easy transit to many areas (Zoo, Manly Beach, Harbor Bridge, Opera House, etc etc.).
Have fun,
Cor
ps: we'll be going to the Bilikiki about a month after you again 