While many calculus texts contain a section on hyperbolic functions, this section is often labeled as optional. Many calculus teachers tend to skip or go light on the teaching of hyperbolic functions due to their complexity and inconvenient derivatives and integrals. An interesting Calculus III project would be to study these functions and present your findings. Interesting aspects about these functions are their relationships to the standard trigonometric functions, their derivatives and their applications.
Much of what is taught in advanced calculus courses can be applied to the real world in terms of optimization. For example, setting a function's derivative equal to zero will yield critical values that lead to the optimal solutions of the original function. An interesting Calculus III project in this area would be to find some real-world uses of such processes. You can analyze the functions that describe real-world phenomena, find the optimal values of such functions and interpret the solutions in terms of how to generate the best results.
Sir Isaac Newton created calculus, but he did so not for the sake of mathematics but for physics. A thought-provoking subject for your calculus project would be to re-relate calculus to the world of physics. By Calculus III, you should have learned concepts such as Newton's Method and Taylor's Theorem, both of which are used to approximate the functions of complex physical phenomena. Many of these advanced concepts have wide applications in the world of physics, from Newtonian mechanics to optics. You can create a project that demonstrates how the theory of calculus is applied in the physical world.
There are many paradoxes in calculus that do not get mentioned in a Calculus III course but can be solved with Calculus III methods. For example, you can create an interesting project that presents some of these paradoxes and their solutions. Some examples of these paradoxes are Zeno's Paradox, the convergence and divergence of strange sequences (e.g., the sequence {-1, 1, -1, 1, ...} and equations that seem to allow for division by zero.