What is the name of the hypothetical particle that mediates gravity?

The graviton, and hypothetical is doing a great deal of work. The other three forces have confirmed carrier particles - photons, gluons, W and Z bosons - and gravity is described by general relativity as curved spacetime rather than by particle exchange at all. The graviton is what a quantum theory of gravity would need, and there is no such theory.

Detecting one directly is probably impossible. Gravity is so weak that a detector able to catch a single graviton would, by one well-known estimate, need to be the mass of Jupiter orbiting a neutron star, and would be destroyed by neutrinos long before it registered anything. LIGO detects gravitational waves - ripples in spacetime - which is not the same as detecting the quantum.

The gap between general relativity and quantum mechanics is the largest unfinished business in physics. Each is astonishingly well confirmed in its own domain and they contradict each other where both should apply - inside black holes and at the first instant of the universe. String theory and loop quantum gravity are the main attempts, and neither has produced a testable prediction that has been checked, which is an uncomfortable position for a field that prides itself on being empirical.