Your friend moved out without notice. Even following a spat with the landlord, and his being unreasonable as to her having overnight guests, that could potentially open her up to responsibility for some rent. As this was a month-to-month tenancy, she would ordinarily be expected to give a month's notice. Presumably the "Fair Housing Authority" suggested that she get the landlord to issue some form of documentation indicating that she was evicted, as otherwise it would be difficult to make a claim for the money the landlord retained.

She may want to discuss her situation with a local lawyer, so she can get a clear opinion as to her rights and remedies.