Deliver a laser pistol to Eldridge

Speak to Jules, and ask about the Salvatores. Ask where they get their "lightbringers" (aka, laser pistols), and he'll say he doesn't know. Go see Eldridge and ask if the Salvatores buy the pistols from him. He'll ask the player to retrieve him one to study, and will pay handsomely for it.

Although the player promises a pistol from the Salvatores, pretty much any basic laser pistol will do. If the player serves Louis Salvatore loyally, they can eventually ask him for a laser pistol before his third mission for them, which they can then deliver to Eldridge. Of course, outright killing the Salvatores and taking their laser pistols works just fine too.