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...the fact that I applied to a math research program in my application.
I just got rejected today.
My summer plans will probably be kind of lame. (Part time job and studying for the LSAT).
Is it worth adding the fact that I did try to strive for something greater this summer or should I just really play on the fact that I was productive but not extraordinary this summer?

Why do you feel compelled to mention anything about your summer? I assume you aren't starting law school this fall (since you will be studying for the LSAT). Law schools won't look at doing "nothing" during summers at undergrad negatively.

The idea of mentioning that you applied for something but were rejected seems rather absurd.