There’s some wonderful hiking around Mjóifjörður. For a fee, the folks at Sólbrekka can ferry you across the fjord, from where it’s a four-hour hike to Neskaupstaður, or you can climb over northern mountains to reach Seyðisfjörður on a six- to seven-hour trek (it's best to hike in late summer, from mid-July or later, when the high-altitude snow has finally melted).
From the head of Mjóifjörður it's 12km to the settlement at Brekkuþorp (often labelled Iceland’s smallest village), then the road continues 14km east to the Dalatangi light, Iceland's first lighthouse, from 1895 (next to it is the 'modern' one, dating from 1908 and still in use). Beyond Brekkuþorp a 4WD is advisable.
On the north side of the fjord at Brekkuþorp, Sólbrekka is the one and only place to stay around here, and it’s a welcome sight for hikers.
Coffee and light snacks are available at Sólbrekka Cafe, and breakfast and dinner can be arranged for overnight guests. Bring supplies from Egilsstaðir or Reyðarfjörður.