What do we learn of the real actual Hester Prynne in the surviving papers?

There really isn't too much to say about Hester in Custom House. The narrator stumbles upon the Scarlet Letter and an account of Hester Prynne. He gives a rough outline of Hester's life without any real detail.

"She had flourished during a period between the early days of Massachusetts and the close of the seventeenth century. Aged persons, alive in the time of Mr. Surveyor Pue, and from whose oral testimony he had made up his narrative, remembered her, in their youth, as a very old, but not decrepit woman, of a stately and solemn aspect. It had been her habit, from an almost immemorial date, to go about the country as a kind of voluntary nurse, and doing whatever miscellaneous good she might; taking upon herself, likewise, to give advice in all matters, especially those of the heart; by which means, as a person of such propensities inevitably must, she gained from many people the reverence due to an angel, but, I should imagine, was looked upon by others as an intruder and a nuisance." (Custom House)

Suffice to say the speaker was impressed enough to pen his own version of the story.