Here are a list of ideas for projects, along with a source.
I advise you to find at least one other reference -- otherwise it becomes tempting to essentially copy what's given in one source.
Some of these are more advanced and I have marked them.

I'll add more topics as I think of them.
EC1/EC2 refers to "Enumerative Combinatorics vol. 1" (second ed.) and "Enumerative Combinatorics vol. 2" written by Richard Stanley.

• Cyclic sieving phenomenon (Sagan, 6.6 and this paper)
• Combinatorial reciprocity (Chapters 1, 2 of this book for example)
• Schubert decomposition of flag varieties (EC1, 1.10)
  This is a very big topic, but the combinatorial formulas here are self-contained and have a few natural extensions that are still within the scope of the class.
• Gessel-Viennot lemma (EC1, 2.7; Sagan, 2.5)
• Hyperplane arrangements (EC1, 3.11 and also here)
  This is a really big topic, so you should stick to just a few main points.
• Ehrhart polynomials (EC1, 4.6.2)
• Combinatorial proofs of Lagrange inversion (EC2, 5.4)
  There is also a q-version which is interesting.
• Matrix-tree theorem (EC2, 5.6; Sagan, 2.6)
• Basic theory of algebraic generating functions (EC2, 6.1-6.3)
  Assumes familiarity with field extensions, for example Math 100C.
• Catalan objects (EC2, Exercise 6.19)

The following topics are also good, but will probably require more effort on your part to find accessible sources.

• Euler's pentagonal number theorem (wiki) and other partition identities (notes from Wilf)
• Eulerian polynomials and q-analogues (wiki)
• Finite difference calculus (wiki)
• Faà di Bruno's formula for nth derivative of a composition of functions (wikipedia)
• Regular languages (wikipedia)
• Counting tilings
  I am less certain about an introductory source, but one place to start could be here.
  Warning: It might be a lot more work than the other topics.
• Combinatorial species (wikipedia)
  If you have some familiarity with category theory.