Lots of travellers stay at the campground on the main road into town. There are 11 decent-value cottages, sleeping up to six; some have private toilet, but all use the amenities block for showers. There's also a playground and laundry, and some camping gear is sold at the reception. The downsides: very few showers for the capacity, and no wi-fi.
Höfn Camping & Cottages
Southeast Iceland