There was a mistake made.....but not by God.
I do not personally believe that this is a case of God saying to Himself, "Oops, I wish I hadn't done that!".
If that were the case here, (and other places where it says He "regretted"), then because He is God, He could have and would have wiped out the action and started over. And He didn't do that.
God never wanted Saul to be king in the first place. In fact, He never wanted his people to have a human king. The people asked for a king. They basically said, "Hey! All of these other countries have a king, why can't we have a king!?!?!?"
It wasn' so much that they wanted a king, but was more likely that they didn't want God to be sovereign over them.
God spoke to Samuel and said, "Tell them that if they have a king that he will force their sons to be soldiers and laborers for his profit and he will force their daughters to be laborers and that he will take their vineyards and properties and he will force a tax on them. Tell them that having a king won't be nearly the wonderful life that they think it is going to be."
The people didn't care. So God gave them what they wanted. Someone who looked good. The movie star "type". Someone who was tall and whose countenance and personality was contrary to what was in his heart.
And the people suffered for it. And it broke God's heart.
Parents suffer when their children suffer. Imagine having a daughter who against your better wishes married a very wealthy and handsome man, but who treated her like trash.
Imagine you had to be a witness to her daily unhappiness and because of her stubborness, she wouldn't come and tell you how miserable she was and how much she should have listened to you in the first place.
You would be in much pain over her pain. You would be aggrevated beyong belief that she didn't listen to you. And it would grieve you to see your grandchildren being raised by this man.
God grieved for his people in that they wouldn't seek his face. God grieved and regretted the fact that this was how his people had turned out and that He had to let them learn from their mistakes about how Saul was not the answer for them.
God regretted that Saul was King. Not in the fact that He made a mistake. God knew before he created the planet Earth that there would be a man named Saul that his people would choose over Him. And it hurt Him even then.
After they reject Him, God gave his children what they wanted. But like a very loving father, who will not force his children to obey him, God allowed his children to make the mistake and eventually learn from it.