Honiara is not a pretty city and not really a cheap place to have a relaxing holiday. The infrastructure is poor, the streets are full of rubbish, and transport is limited.
You need to be patient with how things work there. Arranging tours is not terribly easy if you're a lone traveller. The best places to check out first are Guadalcanal Travel Service and the Visitor Centre as they will help you arrange tours as info on the internet is very limited. The things to do are going out to Savo, Tulagi islands or the Central Province, but it depends on the availability of transport. The diving at Mbonege is excellent. The WW2 tour is so so, most of the time you'll be on the road (full of potholes). If you have time, the boat to the western province (the most beautiful part of the country) leaves on Sunday, gets there late Monday and comes back on Wednesday, so really you need at least 2 weeks and lots of $$ if you wanna have a diving trip. Most other places in the country are only accessible by plane. The people are friendly but it's not advisable to be out at night alone. Restaurants are limited, apart from the expensive hotels (averaging $25/meal). I found a cheap option that's the Chinese restaurant Golden Crown. Fruit at the Central market is also very cheap. Lime Lounge is a popular place among tourists as it's got free wifi, good food and air conditioning.