Hester does penance for her sin by simply remaining in the colony in the first place. She could simply leave, go home or go elsewhere in the colonies, and begin life anew with her daughter, but she chooses to remain. 

Here, she said to herself, had been the scene of her guilt, and here should be the scene of her earthly punishment; and so perchance, the torture of her daily shame would at length purge her soul, and work out another purity than that which she had lost; more saint-like, because the result of martyrdom.

Hester hopes that, by remaining in Boston, the process of atonement through the torture of her shame and public punishment and humiliation will be more complete, so she will acquire a certain kind of purity as a result. She's conscious it is her choice to stay, and she does so with intention and purpose. She feels her daily shame in this place will eventually wipe clean the sin from her soul, and though she can never be innocent of sin again, her choice to remain will bring her a different kind of clarity and goodness.