Link to original question:
http://www.theanswerbank.co.uk/Business-and-Finance/Personal-Finance/Question1500955.html
The rental contract is one solely between the father and the landlord, so the council staff have correctly stated that there's no obligation upon her to pay any rent
directly to the landlord. (It's the father's responsibility to ensure that the rent is paid, either by him or through the benefits which he receives)
However the father is obliged to inform the authorities about the income of other family members living in the house, meaning that (now that the daughter is earning) he will almost certainly suffer a reduction in benefits.
It would therefore be appropriate for the daughter to pay money to her father in return for her staying in the house but any such arrangement is entirely separate to the father's contract with the landlord (and thus not officially regarded as 'rent').