If a friend rented a room from me but moved out to day after she paid her rent, do I have to give her a refund?

Although there wasn't any formal lease, the payment of rent by the tenant and your acceptance of the rent payment means that there is a month-to-month tenancy.  A month-to-month tenancy requires thirty days written notice in order for the tenancy to be terminated.  Since your tenant did not give you thirty days notice, she is liable for the rent for the entire month. However, if you find another tenant before the thirty days are up, you would have to refund the prorated amount for the time from which the new tenant takes possession through the remainder of the month.  For example, if a new tenant moves in on the fifteenth, you would have to refund one half the rent paid by the previous tenant for that month.  You will need to mitigate (minimize) damages (the amount you are claiming you are owed) by making reasonable efforts to find another tenant and cannot allow the rental to remain vacant for the duration of your former tenant's tenancy without making reasonable efforts to find another tenant. Failure to make reasonable efforts to find a new tenant would be a failure to mitigate damages and your damages would be reduced accordingly.  Reasonable efforts to find another tenant would be determined by what other landlords in the area are doing to attract tenants such as posting a sign advertising the rental, advertising in the newspaper, a local rental guide or online, etc.