How come Saul did not remember the name of David's father?
Saul first learned of David, son of Jess when he was looking for a musician in 1 Samuel 16:
18 One of the young men answered, “Behold, I have seen a son of Jess the Bethlehemite, who is skillful in playing, a man of valor, a man of war, prudent in speech, and a man of good presence, and the LORD is with him.” 19 Therefore Saul sent messengers to Jesse and said, “Send me David your son, who is with the sheep.”
In the next chapter, David killed Goliath.
Saul could not remember the name of David's father,
1 Samuel 17:
55 Now when Saul saw David going out against the Philistine, he said to Abner the commander of the army, “Abner, whose son is this young man?”
And Abner said, “By your life, O king, I do not know.”
Abner did not know either. Before this day, David wasn't important enough for Abner to know his father's name.
56 The king said, “You inquire whose son the youth is.”
The king insisted to find out.
57 So when David returned from killing the Philistine, Abner took him and brought him before Saul with the Philistine’s head in his hand. 58 Saul said to him, “Whose son are you, young man?”
Saul had forgotten.
And David answered, “I am the son of your servant Jesse the Bethlehemite.”
How come Saul did not remember the name of David's father?
He just forgot. When he first hired David, Jesse was the father of a musician. Saul didn't think it important to remember the name. Neither did Abner.
Now that David had killed Goliath, Saul was impressed. Jesse was the father of a mighty warrior. Saul wanted to know whom David was from. Now, Saul made it a point to remember the name "Jesse".