In The Lion, the Witch and the Wardobe, the White Witch is intent on getting all four of the children to her house.
"Oh, but if I took you there now," said she, "I shouldn't see your brother and your sisters. I very much want to know your charming relations.
and
"How dare you come alone?" said the Witch in a terrible voice. "Did I not tell you to bring the others with you?"
Why does she need all four? Why not just kill Edmund? Without Edmund, the prophecy couldn't be fulfilled.
We're only seeing part of the story. There's a lot going on under the surface of the story that we know which never gets directly discussed - things like exactly who Aslan is, or the Emperor from Beyond the Sea. Then, of course, there's the Deep Magic which requires Aslan's sacrifice.
So, while it's never spelled out in exact detail, it's likely there is some magical benefit to killing all of them.
Prophecies can be misread. While it is generally believed that all four thrones must be filled by brothers and sisters to fulfill the prophecy, that's not actually spelled out in the wording of the prophecy that we know.
"When Adam's flesh and Adam's bone/Sits at Cair Paravel in throne/The evil time will be over and done."
That Cair Paravel has four thrones suggests that four are needed, but it's not explicit. Safer, then, to eliminate ALL the possible candidates for any of those thrones in one go.
She does. In the end, when it is clear that they cannot catch the the other three, the White Witch does attempt to kill just Edmund:
Chapter Thirteen
...
"Yet it might be better," said the dwarf, "to keep this one for bargaining with."
"Yes! And have him rescued," said the Witch scornfully.
"Then," said the dwarf, "we had better do what we have to do at once."
"I would like to have done it on the Stone Table itself," said the Witch. "That is the proper place. That is where it has always been done before."
...
Edmund found himself being roughly forced to his feet. Then the dwarf set him with his back against a tree and bound him fast. He saw the Witch take off her outer mantle [...]
"Prepare the victim," said the Witch. And the dwarf undid Edmund's collar and folded back his shirt at the neck. Then he took Edmund's hair and pulled his head back so that he had to raise his chin. After that Edmund heard a strange noise - whizz-whizz-whizz. For a moment he couldn't think what it was. Then he realised. It was the sound of a knife being sharpened.
So it's important to read what's going on here - Edmund isn't just going to be casually killed, but ritually sacrificed. Even at this point, when she's given up on catching the others, she doesn't just turn him to stone the way she did with other complications, or have the dwarf throttle him on the spot. She has him bound, removes her mantle, sharpens her stone knife...
It's still about more than just getting rid of a complication. She is preparing for a ritual sacrifice, presumably to draw some kind of advantage. One can only imagine what sort of advantage she might have received from killing all four of the Pevensies on the Stone Table itself.
If you're going to do a job, might as well do it properly. Yes, killing Edmund would prevent all four from becoming kings and queens as prophesied. But the remaining three would still be able to be crowned, thus fulfilling the prophecy as nearly as possible at that point. Perhaps they would even have found a fourth person, not their brother, to fulfill the prophecy completely.
Trying to stop a prophecy from being fulfilled never really works. The White Witch was trying to do something which is sort of impossible, so don't be too surprised if her methods don't completely make sense. But if you want to prevent the outcome of a prophecy, it's probably best to do so as completely as possible, by killing all the people involved rather than just one.
Plus, killing someone's brother tends to make them your enemy. The White Witch didn't need three people roaming around with reason to hate her who were potentially prophesied to replace her as Queen. Much safer to get rid of them all.
Put yourself in her shoes. You're an evil megalomaniac who's been told that four specific people may cause you to be removed from power. Would you settle for killing a single one, or would you try to get rid of them all?
The Witch likely thought she could subvert them all, and thus beat the prophecy. I mean, Edmund sells out for a box of sweets. She likely thought the other three could probably be just as easily bought off. Here's your heart's desire, now do my bidding (fits into the Witch = Satan part of the allegory).
Another angle is she really wants Aslan. I'll spoiler this part, since it's partially from The Magician's Nephew
The Witch has been in Narnia since its creation (she's literally there watching when Aslan makes it). This is why she knows Aslan and the power he holds. With the children in her grasp, she can get rid of her true nemesis and be done with the person who made the prophecy in the first place.
Even if all four thrones must be filled, the prophecy says nothing of Pevences, just Adam's bone. Any human would suffice.
The Witch makes first contact with Edmund. She needed to know where a human could have come from. Edmund tells her that there are two humans that know of the passage to Narnia. Him and Lucy.
If she kills Edmund that leaves one remaining human with knowledge of Narnia, who probably will be more insistent on returning to search for her brother. She needs to create two more witnesses to eliminate all witnesses.
The witch doesn't know they will themselves fulfil the prophecy and is probably more concerned about other humans coming and completing the prophecy than just four young children.
One possibility is that Aslan is....Aslan. The Chronicles of Narnia is a Christian allegory with Aslan fulfilling the role of the triune God of Christianity (in The Lion, Witch, and The Wardrobe he is primarily representing God, The Son).
Does Aslan want Edmund to die? No, therefore Aslan's will is carried out and Edmund is not killed by the White Witch. In Christian theology, God's will, purpose, and plan is always fulfilled. Even when it looks like evil is getting their way (the story of Job or Joseph), God is restricting their actions and/or using them for His purposes.
The prophecy only requires Adam's flesh and Adam's bone to sit at Cair Paravel. The White Witch had to get rid of all of that, so that there would be none of Adam's flesh and bone to sit at the throne.
"When Adam's flesh and Adam's bone/Sits at Cair Paravel in throne/The evil time will be over and done."
