If she overstays how long rent is paid for, he will have to file an eviction action to remove her; assume that if she overstays, he will lose between 1 and 3 more months of rent as he gets her out, plus incur court costs (and possibly legal fees, if he hires an attorney). If she damages the apartment, he will bear the repair cost himself, since there is no security deposit. As an illegal alien without job or the legal right to work (no green card), it is effectively impossible to sue her for any of these amounts. Therefore, he is risking potentially considerable loss of money; it would be wiser to let her stay with her family.