The people of Israel here do a curious thing -- they see a problem and in order to solve it, they ask for it to get worse. Don't ask me why.
The problem is that Samuel's sons, whom Samuel has named judges to follow him, are not very good at their job. This really shouldn't surprise anyone. The "judges" we read about in their book were not exactly like our own judges -- they were religious leaders who were thought to be gifted by God's spirit to lead the people in a time of crisis. The people would fall away from following God and trouble would come. God would raise up a leader or "judge" to help them and then things would get better. Then they would fall away and the cycle would start over.
This kind of leadership couldn't be inherited. Being the son or daughter of a judge was no guarantee that a person would receive the same spirit, and so we have men like Samuel's sons, who are only magistrate-type judges and not very good ones at that, since they are corrupt.
But when the people ask Samuel for a king, they are asking for exactly the same kind of situation, only worse. There's no guarantee that the eldest son of a good king will himself be a good king -- there's only the guarantee that he's going to be king. If illness or intrigue or war intervene, he may not be king at all, but that's also no guarantee that there will be a good king on the throne. If you have a good king, Samuel says, that might be better than a bad king, but the king will still draft your sons, use your daughters in arranged marriages to his own political ends, and tax you, and you won't be able to stop him. Kings in that culture didn't have Parliaments or other restrictions on their power. Even though the Israelite kings would have the Law of Moses, they could (and often did) ignore it if they wanted to.
The people stick to their guns. They want a king like other nations have. And that's where we see something interesting crop up. The people weren't necessarily asking to have a monarchy begun. They had a king: God. God was the ruler of the people of Israel. God had given them their law and, though his spirit, guided them in following it. The Israelites asked for a change of kings, from having God as their sovereign to having a human being.
Does that make a difference? Well, several, but I wanted to point to one of them. When God was their sovereign, the Israelites had the Law of Moses to follow. But they had to interpret it for themselves sometimes, because some things weren't as clear-cut as others. They could always seek God's guidance, but we know that while God responds to our prayers that response is rarely the words in the sky declarative sentence we want. They had to develop, with God's help, their ability to discern and judge. They had to mature, if you like.
A human king, however, can always have a definitive answer. "Lord, I know I am supposed to honor my father and my mother, but my dad took off when I was one and is back asking me for a loan. What should I do?" God's response is not likely to be as clear as a king's: "You're right, he's a bum. Kick him to the curb." That may sound great, but the problem is that when someone answers every question for us, we never learn how to develop our own answers. What do you call a being whose every action is dictated by your will? A pet, that's what! And we are not God's pets!
God desires our growth, our maturing in our lives of faith. He meets us where we are but he wants us to get better, and he leads us in doing so. So he doesn't cover all the bases and nail down every detail -- if he did we would never grow into what he has in mind for us to become. No earthly king, teacher or ruler can do this; only our Lord and savior. And it is what he desires for us more than anything.