5 weeks -- Cooks or Solomons?

I'm a 33-year-old male looking to unplug for 5-6 weeks in the South Pacific. Mainly interested in finding the best beaches to chill on and do some hiking, maybe ride a moped, Don't care at all about bars, partying. On a limited budget. Would like to get away from over-touristed areas too, maybe even sleep out under the stars on a deserted beach or in a hut. Will be carrying travelling lightly, but would like to find a couple sweet places to chill for a while.

Where would you recommend going? And why?

Thanks muchos.... and happy trails.

I haven't yet been to the Cooks, but I think with the time you have, the Solomons give you many more options, from laid back resorts to village stays.
Go to the address in my signature line below this post, and look at all the things you could do.

I don't think I've ever seen a moped anywhere in Solomon Islands. However, you can go lots of places by boat, or in the backs of trucks, or even on a trailer behind a tractor.

"maybe even sleep out under the stars on a deserted beach...."

It can be done but be sure you're well immunised.

Camping is not allowed in the Cooks. You will find mopeds. Deserted islands where you can camp and moped rentals generally don't go together. You could rent the hut on beautiful One Foot Island off Aitutaki but it will be at least mid-range in cost and you will have daily tour boat visits. While I would not consider the Cooks as over-touristed they (Rarotonga and even Aitutaki) might be too busy for your taste. Look at the Ha'apai in Tonga - especially Uoleva where you can live in a hut on the beach.

Camping is not usually encouraged in the Solomons either, but you can stay in a village leaf hut or guest house by arranging with the chief. (It is not worth carrying camping gear, except a mat, if you are travelling by local ship).
If you go there, stay in one of the church Rest Houses when you arrive in Honiara (don't forget to book in advance, they are very popular with locals.)
If you do that, you will get to talk to local people, and may get invited to their home villages, or get lots of other travel ideas.

I don´t know if it´s relevant, but isn´t malaria an issue on the Solomons?

Yes, malaria is a big issue in Solomons. If you go, be sure to go prepared.