New Ireland/PNG/Louisiade on the cheap?

Is there a way to travel to one of these places or anywhere else in Melanesia that is really cheap? I plan to snorkel and hang out on beaches most of the time, enjoying the culture and people and seafood.

Basically, there's no way to travel in the entire South Pacific really cheaply.

The cheapest part of Melanesia would have to be Indonesian West Papua, and even that is much more expensive than western Indonesia.
PNG doesn't have to be ultra-expensive, but even budget accomodation costs significantly more than in Asia (or West Papua), and just reaching New Ireland costs quite a bit of money.

Internal boat travel in Solomon Islands is relatively cheap, but you get what you pay for, a couple of square feet of deck to put your sleeping mat.
How much research have you done about Melanesia? The islands aren't all white sandy beaches you can lie on - in fact there are very few of these. If you go to out of the way places, look for village stays.