If a Museum is a place where mathematics meets people of every kind, it is important to let them think that our discipline is useful, and is not just a game. I advocate to display applications of good mathematics in the everyday life. Cryptography has been evoqued before and I voted +1 for this answer. Let me add a few others :
- Radon transform, with application to tomography, and therefore to medical diagnosis.
- QR algorithm, with application to searching on the web (Google page rank algorithm).
- Dynamical systems, saddle points and their application to the launch of spacecrafts away from the ecliptic.
I have not been involved in the elaboration of any mathematical exhibition, but I am convinced that if these topics have been successfully used by non-mathematician, they can be explained to a non-scientific audience. I except that they contribute to a positive judgement of mathematics by the population.