Thane of Eastmarch

The Dragonborn can gain the title of Thane of Eastmarch.

Objectives

 * Befriend the Jarl of Eastmarch.
 * Assist five citizens of Eastmarch.
 * purchase Hjerim
 * Return to the Jarl.

Imperial Legion Questline
After completing the "Reunification of Skyrim" quest, Brunwulf Free-Winter will become Jarl of Windhelm. The Dragonborn must speak to Brunwulf, and Brunwulf will offer the title of Thane contingent on them helping five people of Windhelm and purchasing Hjerim from Brunwulf's steward. After these two objectives have been completed, Brunwulf will grant the Dragonborn the title of Thane, give him a random leveled weapon of Eastmarch, and appoint a housecarl to serve them.

Stormcloak questline
After progressing through the questline "Liberation of Skyrim": Liberate Falkreath Hold, the Dragonborn will receive the option to become a Thane. This is done by selecting the "My Jarl, is there anything further you need?" conversation option. Ulfric Stormcloak will respond that there is a place in his court for a new Thane, but the title can only be given to one known throughout the hold and that owns property in Windhelm.

Regardless of the questline followed, there are two requirements to becoming Thane of Eastmarch:

There are several easy ways to assist the people of Eastmarch: While completing "Blood on the Ice" is not a prerequisite of becoming a Thane of Eastmarch, the quest will automatically start upon the purchase of Hjerim. Also, if on the Legion's side in the Civil War, Brunwulf's bandit quest must be done to become a Thane anyway.
 * Purchase Hjerim from Jorleif for 12,000.
 * Assist five citizens of Eastmarch.