If you want to go camping in Iceland, you could consider renting a campervan as opposed to brining a tent. Especially if you are planning to go in the spring or fall, that would be the best option because it will keep you a bit more sheltered from the elements.
I absolutely loved camping in this campervan (Go LITE camper). It is very basic and sleeps 1-3 people, although I would say 2 is the max if you want to keep it comfortable.
Just remember you are not allowed to camp outside of camp sites, so make sure you know where the camp sites are (and calculate those costs as well when planning your trip).
You'll get a 5% discount if you use my link here.