What to do if my mother’s landlord is trying to evict her but she never received notice?

The first and most important issue is--why is the landlord trying to evict her? New Jersey has a law called the Anti-Eviction act which limits the reasons a tenant may be evicted--and just because the landlord wants her out (or even because a lease expired) is *not* a legal reason to evict a tenant. Your mother may only be evicted for a reason set out in the Anti-eviction Act.
If the issue is nonpayment of rent, no notice is necessary: a landlord may simply file an eviction action in court (and your mother could avoid eviction by paying the past due amount).
If the reason is that your mother allegedly violated the lease or engaged in disorderly conduct (disturbing other tenants), there would have to be two notices: first a notice to cease doing whatever she was doing and then, if she did not, a second notice terminating her tenancy--and the second notice would have to be a month notice. So if this is the alleged reason, the notice is defective and your mother could be not be evicted on a single 3-day notice, whether or not she received it.
The main reasons that could allow a three day notice of eviction (a 3-day notice to quit/termination notice) are: 1) certain criminal acts committed on the property; or 2) intentionally or with gross neglect damaging the landlord's property. If it was one of these reasons, a three-day notice could be valid, but it would have to have been sent properly and would also have to spell out the reason(s) for eviction in detail; if was not properly served or did not inform your mother of the grounds for eviction, the notice would again be defective and she could not be evicted.