I've been asked to figure out how early an employee typically
punches in for work. They want to give her some sort of bonus. I
have the punchclock data in an excel column in the following format:

5/14/07 10:02 AM
5/15/07 9:16 AM
5/17/07 9:28 AM
5/18/07 9:16 AM
5/22/07 9:15 AM

She's supposed to start work at 10:00am every day, and I need to
figure out what time she usually checks in. I've tried a typical
averaging formula, but that gives me the mean time between all of the
days, which is around noon on the 17th.

Here's one way.

Start with the data in column A.

In B1, put
=A1-INT(A1)
and copy down as far as needed. This is the time-of-day for each day's
arrival.

For the average arrival time, use
=AVERAGE(B:B)
and format as time.