If you understand running for fun or a mile run for fun, you can understand this a mile run for time the same way. In short, this for is used for stating the purpose of the action.
I agree with StoneyB in his comment, for time in this context means aiming for the best time or for the shortest time the athlete can do in the given activity, which is running in this case. This for time is in contrast with another kind of running, which is running for distance.
As someone who loves water sports, I'd like to quote this passage from a book I found on Google Books. It has a similar example usage: a 20-minute swim for distance:
[...] To find the category most appropriate for you, you will take a test.
The test is a 20-minute swim for distance. That is, you will see how far you can swim in 20 minutes. The distance you swim in the 20-minute time period will determine which level you will choose.
--Swimming Workouts for Fitness and Training, by Richard Michaels