How does Hester Prynne from The Scarlet Letter decide where she and her daughter Pearl will live their lives ?

In chapter V of The Scarlet Letter, "Hester at her Needle", Hawthorne narrates how Hester makes a decision that nobody saw coming: Rather than fleeing from the village, forgetting her past there, move on, and getting rid of the ridiculous scarlet letter, Hester actually feels that she has a purpose, a reason, to be there in the first place. 

All is summarized in the paragraph which reads,

Here [...] had been the scene of her...

In chapter V of The Scarlet Letter, "Hester at her Needle", Hawthorne narrates how Hester makes a decision that nobody saw coming: Rather than fleeing from the village, forgetting her past there, move on, and getting rid of the ridiculous scarlet letter, Hester actually feels that she has a purpose, a reason, to be there in the first place. 

All is summarized in the paragraph which reads,

Here [...] had been the scene of her guilt, and here should be the scene of her earthly punishment; [...] 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.

Essentially, what this says is that Hester told herself that she will need to pay for the sin that she committed, forever.  She has actually grown with the scarlet letter, after all. The evils that she has seen others commit against her, the hypocrisy of the people surrounding her, and the sanctimonious nature of the villagers are all examples of what Hester has seen, for the first time ever, once she became the bearer of the token. In not so many ways, the letter gave her that insight of the true nature of the human heart that she would have never had learned otherwise.

Therefore, Hester chooses a cottage in the outskirts of the village, not too far, and not too close from the epicenter of everything. It is also noteworthy that the magistrates would still be on the lookout for her and checking on all of her movements. Hester is not entirely free, but this cottage signifies something she can finally call her own. 

It had been built by an earlier settler, and abandoned, because the soil about it was too sterile for cultivation, while its comparative remoteness put it out of the sphere of that social activity which already marked the habits of the emigrants.

While the place attracts a lot of enigmatic attention from the other villagers, and Hester does get bullied, it was still a place far enough for her to find some peace while still accepting the fate that has fallen upon her.