Question

Which has a greater effect on potential energy: mass or height?

Answer 1

Neither does. Their effects are EQUAL.

Potential energy = (MASS) x (gravity) x (HEIGHT)

You should be able to look at this formula and see that the effects of mass and height are equal.  Gravitational potential energy is directly proportional to both of them.

If you want to increase the potential energy of some object, say, by 42%, you can do it either way ... either increase the object's mass by 42%, OR raise the object 42% higher than it is now.  Either of those changes will have the same effect on the object's potential energy.