The girl is Dr. Heidegger's long-dead fiancee. Hawthorne writes that on
"the opposite side of the chamber was . . . the full-length portrait of a young lady. . . . Above half a century ago, Dr. Heidegger had been on the point of marriage with this young lady; but, being affected with some slight disorder, she had swallowed one of her lover's prescriptions, and died on the bridal evening."
The description of the portrait appears in the early paragraphs of the story, because Hawthorne is employing foreshadowing. Readers should speculate that if Dr. Heidegger accidentally killed his fiancee with one of his experimental tonics, then perhaps his friends should not so willingly participate in another one of his experiments.
Additionally, the portrait most likely serves as an impetus for Heidegger's quest for the secret of youth and life. He speculates that if he can find a way to reverse the effects of aging that perhaps he will be able to reverse death and bring back his fiancee.
the portrait is sylvia...his wife...well dead wife.