How can I overcome the fact that I attend a weak undergraduate institution?
It sounds like you have a good start on the most important thing: take advantage of every opportunity at your current program and excel at all of them. Even from a weak undergraduate program, if you can come out with multiple letters saying that you are the strongest student they've seen in X years and have the credentials to back that up, you'll have a good chance at solid graduate programs. 

Your situation means that standardized tests are also even more important. Math is nice because there actually exists a test (math subject GRE) that most schools put significant weight on, so make sure to take it very seriously. Doing well on the math GRE takes away a lot of the uncertainty that admissions committees will have about your application due to the institution.

Assuming that you don't transfer, some other options:


Get to know faculty and spend that extra time doing research. Proven research experience can make up for a lot of coursework, and you'll hopefully get a much stronger recommendation letter as well. This is important to do no matter what kind of institution you're at, but is probably extra important at weaker programs where your application can't rely on reputation and coursework.
Consider graduating early and getting a Master's degree at a strong program. This isn't always financially feasible, but if it is then it's an option to strengthen your coursework and resume. Availability of strong master's programs can depend on the field, but it's worth looking into.
Graduate early and get a job doing research. Again, availability may depend on field (I hear that these aren't common in math), but if you can get the right job it can be a great boon to a resume and help generate another good recommendation. Bonus points if you can find a job near a good university and take a couple classes on the side.