Living costs in Honiaroa for an extended period

Approximately how much should I budget for per day for food and other living costs? Hotel accommodation is covered, but not food etc.

I couldn't find reference to internet access being provided by any of the few hotels. Is this service available to laptop users?

Thank you

Depends on where you are staying. The expensive hotels all have some form of access (maybe for a fee.) You can get a WiFi card from Solomon Telekom.
Otherwise, the local internet cafes (several in the Anthony Saru (6 story) building in the middle of town) are relatively cheap.
Places like Chester House do not even consider offering internet access - it would be very much in the too hard basket for them - they have enough trouble responding to their own emails much of the time.
Is your accommodation being provided for the period you are there, or do you have to find your own? If you need to find it, you could consider booking the self-contained flat at St Agnes Rest House.
If you have self-catering facilities, and like sweet potato or rice, you can live fairly cheaply. However, if you are relying on imported food, it is relatively expensive.
$7 SBD is roughly $1 Australian.
In SBD a tin of tuna costs $5 or less, 10kg bag of rice $100, you can get local fruit and veges very cheaply at the local market.
If you are paying for cooked food, try Panatina Plaza or the food stalls in the NPF Plaza or Nicky's near the King Solomon Hotel. Expect to pay $50 to $100 for a plate of cooked food, $10 for local fish and chips.

I just looked at you profile - if you are a sailor, consider joining the Point Cruz Yacht Club for the duration of your stay (between the Mendana Hotel and the main wharf). They do quite reasonable evening meals (when the other eating places I suggested are closed).
Also, try the pizza at the Raintree cafe, out at White River, on the western outskirts of town.

copie out of my expenses list . it was february march this year. I did sometimes cooking, but mostly eat out, but places where locals eat:
prices in solomon dollar,
one hour internet- 20 dollar
1 sprite tin- 10 dollar
one drinking coconut- 2 dollar
coffee- 18 dollar
one night chester house- 200 dollar
chips- 25
prepared food at market, food store 5 - 15
making own food with vegetable of market: 10 -15 dollar
newspaper 5 dollar

Note, the eating places I suggested are frequented by locals. The most popular expat place is the Lime Lounge, next to the ANZ bank. It's big advantage is air conditioning and toilets, but there are also toilets at Panatina Plaza (a little out of town, popular with both expats and locals, and cheaper than the Lime Lounge.
There is also the El Shaddai cafe, upstairs in the Tongs Building, behind Westpac, and Amy's, a series of take away cafes with outdoor seating, beside Westpac.