John Gill Commentary 1 Samuel 24:21

"Swear now therefore unto me by Jehovah, that thou wilt not cut off my seed after me, and that thou wilt not destroy my name out of my father`s house." — 1 Samuel 24:21 (ASV)

Swear now therefore to me by the Lord
By the Word of the Lord, as the, Targum:

that you will not cut off my seed after me ;
as was usually done in despotic governments in the eastern countries, and is at this day, when one is advanced to the throne of another, by whom issue is left, who may be rivals and competitors with him:

and that you will not destroy my name out of your father's house ;
by cutting off his seed, the same thing in different words repeated, for the confirmation of it; children bear the names of their fathers, and by them their memory is perpetuated, and cutting off them is destroying the name of their parents.