General:
* Offspring
* Progeny
* Issue
* Heir
* Successor
* Posterity (referring to all future generations)
More specific:
* Child
* Grandchild
* Great-grandchild (or other specific family relationships)
Figurative:
* Legacy (referring to the impact or influence of an ancestor)
* Inheritor (referring to someone who takes over something from an ancestor)
The best synonym depends on the specific meaning you want to convey.