Move the boxes out of the way to access the back room and interact with the letters on the table. See above image for reference.
Talk to the Postman again, and he’ll give you a letter for the old woman.
Now return to the lady in the lighthouse, choosing either of the two.
Give her the letter and perpetuate the lie – This will complete this quest without reward.
Tell her that her lover is dead – This will complete this quest with 1000 gold.