This expression probably comes from the fact that the elephant has the biggest brain of all land animals.  Many people believe that the heavier the brain, the better the memory.

Elephants do have a very good memory.  They are able to remember the entire map of their home range in their head, which is  a mental map of their entire home range, which can be an area as large as 1,000 square miles (an area the size of Rhode Island).  

An an example of how they use their memory, elephants travel in packs.   when the group gets too big, the oldest daughter breaks off to start her own group.  However, she will never forget her first group.  One researcher witnessed a mother and daughter elephant recognizing each other after 23 years of separation.

On top of this, elephants seem to also remember their relationships with dozens, perhaps hundreds of other elephants, some which they only see once in a while. They also have an impressive memory for watering holes and places where food is located.

Although it may not be true than an elephant "never forgets", after all, even elephants must forget some things sometimes (just as humans do), they are recognized for having extremely good memories, perhaps even better than the average human.