Somewhat depending on what you consider to be a C file, that is a highly non-trivial task! I don't think the compiler test would satisfy me because it would generate too many false negatives and maybe even some false positives. I've written plenty of files that I would consider C (or C++) files, but which a compiler would fail to compile, either because of missing header files or because of syntax errors.

In like fashion probing the file for interesting common character sequences is likely to be rather error prone.

What is your friend's use case? Often if you focus just on solving the problem for the use case you can simplify the solution and end up with something that is robust for the task at hand. For example just matching files that have a .c extension may be sufficient and is very easy, fast to implement and very fast executing.