Höfn can hardly be praised for architectural beauty, but a walk around the harbour can be nice and there is a bird reserve south of the harbour with good walkways. Most of the sights in the area are in fact in the nature outside Höfn itself. Nevertheless, the town has a few museums that can be of interest:
Collecting rocks is a popular hobby in East Iceland. This private collection of rocks was recently opened in Höfn's old swimming pool. Rather odd, but definitely interesting for geology enthusiasts.
Ice Land Glacier Exhibition
An exhibition dedicated to life in close proximity to Vatnajökull.
Organized tours with some of the many tour companies based in Höfn may be the easiest way to get around the area, and the safest way to explore the glaciers.
The area surrounding Höfn has some of the most stunning nature in Iceland. The lowland area a narrow band of floodplains between the sea and the glacier-topped mountains, where large glacial rivers are still relatively untamed. These floodplains are of a sort called sandur - the word is Icelandic for sand but has been adopted as the international scientific name for the sandy floodplains of glacial rivers found almost exclusively in Iceland and Svalbard. The largest such sandur is Skeiðarársandur, which wasn't bridged until the 1970s.
The mountains themselves are among the highest in Iceland. Öræfajökull is a sub-glacier of Vatnajökull which contains Iceland's tallest mountain, Hvannadalshnjúkur. In the valleys between the mountains, Vatnajökull has a number of icefalls which are accessible by gravel tracks and for the independent traveler with a good car it's a good idea to search out some of the less popular ones to visit for a private date with nature. Some of the icefalls end in glacial lagoons. The most famous is Jökulsárlón which is literally next to the ring road, 100 km west of Höfn. It's an incredible place where large icebergs break off from the glacier and find their way to the sea.
Finally, although the vast majority of the population in the area is found in Höfn, Þórbergur Þórðarson one of Iceland's most famous writers originally came from the farm Hali. Today a museum in his honor, Þórbergssetur, is located by Hali which is about 75 km west of Höfn.