There are already some good answers here about how to get the sun to stop. I'd like to talk a moment about what the effects of that would be.
Initially, huge gee forces
Right when it stops, as @user535733's answer says, you've got huge inertia problems where anything not fixed to the Earth is suddenly moving at supersonic speeds. Great.
But there are also huge forces involved in the stopping itself that would instantly crush anything on the wrong side of the Earth and, I would imagine, fling everything on the other side completely off. After all, it would likely be dozens or even hundreds of gees to have such a sudden stop, compared to Earth's 1.
Imagine you have an apple and you cover it in honey. We'll imagine the apple is the Earth and the honey is everything on it. Get in the car and hold it out of the window, slowly driving up until you're moving fast. You may need to be in a vacuum to avoid wind from affecting the experiment. Then slam on the brakes as hard as you can. (The author disclaims all responsibility if you take this out of the realm of a hypothetical situation.)
Okay that diverged quickly. BUT my point is that you're going to have some serious problems with all the atmosphere flying off the Earth completely, not just moving at supersonic speeds.
I haven't even talked about what would happen to the Oceans. They'd be all over the place, mostly probably also flying off, but the rest going all over the land.
Your end result is basically tidal locking of the Earth and the Sun. If you heeded what I said about the air being stripped off, you now have an airless planet where one side is mercilessly being bombarded with solar rays, while the other is in shadow. It'll be like the Moon's 28-day Day/Night cycle, but with it never cycling.
If the atmosphere is still there, you might have a similar effect to Venus, where the super-long day means one side gets heated and rushes over to the other, creating huge storms and fast-moving winds. Even if that doesn't happen, you still have huge temperature gradients.
Either way, not great.
Responses to other answers
First off, they're all great ideas. I do, though, want to evaluate the results.
- Zokka's answer about tidal locking I already addressed above.
- Walter Mitty's answer is an ingenious way to get everything attached to the Earth to seem motionless. I'm not sure what would happen to the atmosphere, though. Or if it even matters. If the platform is moving to exactly counter the Earth's rotation, you have what is empirically the same as the tidal locking scenario. Everything on the surface is motionless, so it still has one side constantly in day and the other in night.
- Jasen's, as they said, isn't really under known physics. But if the end result is that it looks like the Sun is motionless, you have, once again, the tidal locking scenario. (If I understand N-body physics correctly, because the Sun and Earth are actually orbiting a center of mass of the system that happens to be inside the sun, there's no difference between the Earth orbiting the Sun and the Sun orbiting the Earth--at least not with our laws of Gravity.)
- Loren's addresses the problem of trying to cushion everything on the Earth so nothing flies off and addresses it ingeniously-- since everything on the Earth is pulled equally, there is no observable gee force on the inhabitants. But you still have the tidal locking problem.
- LSerni and Nepene's answers both involve reflecting or deflecting the light (and, I assume heat) in such a way that it appears that the sun is steady when it isn't. While yes, this means you don't have insane gee forces slowing down the Earth, but since all the light and heat is still redirected to one side of the Earth, you have the massive overheating of that side in the same way as the Tidal Locking scenario.
- user535733's answer I addressed. I mostly just clarified that all the atmosphere and things would likely fly off the Earth because of inertia and would like to point out the effects afterward from tidal locking.
Tidally locking the Earth probably isn't a good idea in a hard science world, no matter how you do it.
My suggestion: Take one of the solutions that slow down the earth without causing problems for the inhabitants. I personally like Loren's where you have gravity projectors. Then you need some sort of heat distribution system. If you've already got a deflection system like in LSerni or Nepene's answers, you can use it to regulate how heat is equitably distributed around the planet. You'd probably want some sort of mirror that is permeable to light but reflective to heat. I don't know if we have something like that already-- it's worth a google. You can use those to distribute the heat and keep the light going right on its normal path.
It's complicated, but I'm sure you can figure out a way to work it in; people like you always wow me with your creativity. ;)
I hope this helps.