Q:

What is a word describing a “long-lasting memory” of someone? 

What noun in English is the best for describing the essence of a long-lasting memory that some person (e.g. spiritual leader or famous musician) leaves after himself?  Is it perpetuation, or commemoration, or maybe heritage, or record, or some other word?  I know the word notoriety meaning ill fame, but I don’t know any word for the good one. 

A:

Legacy

something transmitted by or received from an ancestor or predecessor or from the past (m-w)

"To leave a legacy" is a common way to refer to an individual having a great impact on future generations.   Unqualified it connotes a positive impact, but it can be modified with a negative term (like "legacy of conflict") to refer to a negative impact as well. 

A:

"immortality" comes to mind.  In addition to "the quality or state of being immortal" it also means:

enduring fame.  TFD, "lasting fame" MW, the quality or state of someone that will never be forgotten. 

He found immortality through his films. 
Some people want to achieve immortality through their works. 

A:

"indelible" meaning something that cannot be erased (by the passage of time.) 
A synonom would be "ineffaceable"  (edit - which is unrelated to "ineffable" .)