What are my rights regarding a noisy neighbor?

As a practical matter, your mother may be best off waiting until the lease is up and relocating; that's because the best you could most likely do in any event is terminate her lease and move out early, but doing so will (1) take time--probably at least two or three months; (2) be risky--if done incorrectly, she could end up liable to the landlord; and (3) be costly--even if she is in the right about terminating her lease, the landlord may try to either withhold her security deposit or sue her, forcing her to go to court.

If she wants to try and take action, here's the theory: all tenants have a "right of quiet enjoyment"--the right to use their premises without undue or unreasonable disturbance. The landlord is obligated to take reasonable steps to ensure a tenant's quiet enjoyment. Since in NJ, a landlord may legally evict a tenant who disturbs other tenants after notice to stop, the landlord *should* provide the noisy neighbor a warning to change his behavorior or face eviction. If the landlord does not take steps to fix this problem, that could provide grounds to terminate the lease early without penalty.

However, first the landlord must be given a "reasonable" time to take action (i.e. warn the other tenant) and the the tenant must be given at least some time to change his behavior--that's why it could take weeks or months before  you know if the landlord has failed to take action, and your mother could move. Second, the condition must be serious enough to significantly disturb the peace of the average tenant--so if it's just that it's particularly annoying to your mother, but another person could put up with it, then she would not have grounds to move out and would be liable for the remaining balance of her rent.