My mother had no choice but to give up the diamond ring.
Literal
Mother [topic-は] diamond [possessive-の] ring [object-を] let-go other-than was-not
手放す他なかった = 'there was nothing other than letting go' = 'had no choice but to part with.' ~他ない (ほかない) expresses lack of alternatives. 手放す = to let go of / part with.