Is it permissible to give the money designated for the expiation of an oath to a neighbor whose house burned down instead of to the mosque, and is it obligatory to give out the money a second time?

The question is vague, but if the mother had given the expiation (kaffarah) to be placed in the designated places, then you have disobeyed her command and become a transgressor. You have committed a fault from two aspects: first, according to the majority of scholars, expiation for an oath (kaffarat al-yameen) is not valid when paid in cash; second, if your neighbor is not poor or needy, she is not considered a valid recipient for expiation merely because her house burned down. Even if she were needy, the expiation for an oath is only valid when given to ten needy persons.

Based on this, what was done is not sufficient, and the mother is obliged to pay another expiation. You must return the equivalent amount of money you took because you were the cause of its spoilage.