Legal Question in Landlord & Tenant Law in New York

yep. security deposits may be newly required and/or increased with new leases. Landlords have the right to require leases or they may initiate eviction proceedings. Depending on the type of building (rent controlled or rent stabilized? or under other regulatory scheme?) she may be protected from an excessive increase.

As for her current security deposit, so long as she has some proof of that deposit (in lease, receipt, cancelled "cashed" check) then that is to be considered her money. She may still have to pay an increase, but it is also likely that the landlord might have been required to hold it in an interest-bearing bank account- so it may be worth more now than the actual amount she paid. If the new owner has no knowledge of the previous security deposit, and your mom has proof, then that is too bad for the new owner (especially if they are supposed to have it in an interest bearing account).