Reddit reviews Analysis with Ultrasmall Numbers (Textbooks in Mathematics)
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Some elementary geometry of differentiable curves (check out Gibson's book) would probably be a cool project. This is how differential geometry began, which has applications in physics (the book by Gibson talks about the geometry of trajectories).
Another idea: infinitesimals, the foundations of calculus, and nonstandard calculus (check out Henle or Hrbacek). This also has a historical component, which might make it more interesting. You can substitute nonstandard calculus into most applications of standard calculus, so just use one of those for an application.
A project on a specific function with interesting properties might also be a good idea. For example, the Gamma function, the Lambert W function, or even the Riemann Zeta function to name a few. These pop up in a lot of place in and outside math.
I think the fractional calculus has some applications in quantum physics, but that would probably be hard to talk about.