Now David said, “Is there still anyone who is left of the house of Saul, that I may show him kindness for Jonathan’s sake?” 2 Samuel 9:1

It was a common practice for a new lineage of a king to rid himself of the old. As long as one of them lived the chance of an attempt to regain the kingdom existed.
Certainly, Mephibosheth has a legitimate claim to the throne as the grandson of King Saul.
But David does not seek the descendants of Saul for punishment but for kindness.
Mephibosheth son of Jonathan, son of Saul was brought to the king’s palace were David offers him life and security.
He will eat at the king’s table like David’s other sons.

A person after God’s own heart is one that extends mercy and kindness. Such a person looks for opportunities to give life more abundantly rather than death and destruction.
I certainly realize many people will bite the hand that feeds them. If they do then the Lord judge between them and me. I have shown good and they returned evil.

Repay no one evil for evil. Have regard for good things in the sight of all men. If it is possible, as much as depends on you, live peaceably with all men. Beloved, do not avenge yourselves, but rather give place to wrath; for it is written, “Vengeance is Mine, I will repay,” says the Lord. Therefore “If your enemy is hungry, feed him; If he is thirsty, give him a drink; For in so doing you will heap coals of fire on his head.”
Do not be overcome by evil, but overcome evil with good. Rom 12:17-21