The term Hilbert space refers to a complex vector space on which is defined an inner product and which is complete in the norm defined by this inner product.
They are named after the German Mathematician David Hilbert

This term was first coined by John Von Neumann in a seminar he gave. Apparently Hilbert was in the audience and when Von Neumann had finished, Hilbert asked him:"Well Johnny, all this is very fine but what are these Hilbert spaces you are talking about!"