The jarl grants the title of thane and all that comes with it.The jarl asks you to purchase a house in the city if the hold is one of the five with major cities: Whiterun (Whiterun), The Rift (Riften), Eastmarch (Windhelm), Haafingar (Solitude), and The Reach (Markarth).The jarl asks you to demonstrate your value to the hold by assisting the people in the hold, five people in holds with major cities and three people in the other holds.You befriend the jarl through a quest or favors.
The general process for becoming a thane comes down to a relatively general formula with the exception of Whiterun, where it is initially achieved by completing the quest Dragon Rising and does not require the steps below to complete, unless you have joined the Stormcloaks and completed Battle for Whiterun. However, a crime can be pardoned twice if a hold is captured during the civil war and the new jarl appoints the player thane.
