Polya's Induction and Analogy in Mathematics has a chapter on this, along with some great examples. It's not just physical intuition influencing mathematics; it's more a powerful synergy between physical and mathematical intuition. I'll summarize some of it:
Suppose we have two points A and B on the same side of some line L in the plane. What's the shortest path from A to L and then to B? The solution is obvious once we reflect one of the points (and its segment of the path) across L. That solution seems tricky in the abstract, but it's very intuitive if we imagine a reflecting ray of light and think about looking at things in a mirror.
Now suppose A and B are on different sides of L, and a particle moves from A to B, and its speed is different on the two sides of L. What's the shortest path (in time)? (This problem is to a refracting ray of light as the previous one is to a reflecting ray.) It turns out this can be solved by reducing it to a physical problem involving a system of weights and pulleys at equilibrium. I won't try to describe it here, but it might be fun to try to reinvent it.
Now let's take a serious math problem: what plane path minimizes the time an object takes to move from point A (at rest) to point B, assuming constant gravity? (This is the famous "brachistochrone" problem.) By conservation of energy, the speed of the object at a point on the curve depends only on its height (defined relative to its starting point and with respect to the direction of gravity). Thus, we're led to consider light moving in a very particular heterogeneous refracting medium, where the index of refraction depends in a specific way on the height. To find the path taken by light, we simply apply the law of refraction to this medium to obtain a differential equation for the path, which we can then solve.
The interplay between mathematical and physical intuition is very interesting here. The first problem is mathematical, but in trying to solve it, it's natural to draw an analogy to optics. The second problem is suggested by optics, but we solve it by analogy with mechanics. The third problem is basically mechanical, but we solve it by analogy with optics, and we actually use the solution to the second problem!