How would I make a hingeconstraint point to the mouse?

Assuming the hinge is rotating around its X axis (I think that's the only possibility, not sure if that can be changed or not) then the formula would be the inverse tangent of the mouse's CFrame's Z component over its Y component, once you transform the mouse's CFrame to be relative to the hinge's attachment. Complicated? It's easier if you can see it laid out in code:

local relativeVector = hinge.Attachment0.CFrame:ToObjectSpace(mouse.Hit).LookVector
hinge.TargetAngle = math.deg(math.atan2(relativeVector.Z, relativeVector.Y))

If you care to understand the math a little better, here is the math involved.

With the hinge rotating around the X axis, you can see the other axises are positioned like so

When taking the LookVector of the CFrame relative to the hinge, you get a unit vector that represents a direction from the hinge. If you move this direction towards the hinge you can see that it forms a triangle, with the vector itself being the hypotenuse and its components being the sides (Z and Y values). The angle you need to rotate the hinge by is located in this triangle. You have all the sides, so you can find the angle using an inverse trig function.

We're going to use inverse tangent because it allows us to use Roblox's math.atan2 function, which takes into account the signs of both the values that form the ratio. By taking into account both signs, we can make sure our angle is located in the proper quadrant if viewed as a unit circle. Without this, we would get the wrong angle 50% of the time, because there are 2 angles to every tangent value, and the regular math.atan function will give us the same 1 angle every time regardless of which angle we really want. I won't elaborate further on the use of atan2 but if you're interested in understanding it better feel free to let me know.