There are many examples from computing but I guess you're after areas that have mathematical interest in their own right.
I think I good one is the Monte Carlo method. The idea has been around since Buffon's idea of estimating $\pi$. But it became important leading up to, and during the Manhattan Project, when Fermi and Ulam both came up with the method as a way to make impossible seeming integrals over high dimensional spaces tractable. For example, those involved in neutron transport. Today it's in use everywhere from finance to 3D graphics (with Pixar holding the patent on its application to ray-tracing).
There have been all kinds of interesting variations such as Markov chain Monte Carlo and various 'stratification' strategies to increase reliability through 'even' sampling, and Las Vegas algorithms that guarantee correct results. There have been some interesting recent 'pure' applications to mathematics, eg. allowing 'exact' samplings of spaces such as the domino tilings of diamonds. And randomised algorithms are routine in areas like number theory - eg. primality testing.
With hindsight it seems like an obvious idea but I think that at the time the idea of using random numbers to compute a non-random quantity was a pretty big shift of mindset.
The Manhattan Project probably "had a significant impact in society".
