In a story filled with space ships, laser guns, and self-aware computers, who'd have thought an antique brass cannon would be a symbol? But it is, and despite not being mentioned too often, it's a pretty important one.
Prof purchases the cannon while on Earth, and Mannie says it used to be a signal gun used during the days of ship sailing. A signal gun is basically a cannon fired to mark a certain time of day, like high noon. Mannie notes that it "[reeks] of ancient history, pirates, and men 'walking plank'" (18.99)—and this history connects the brass cannon to the idea of rebels, which Mannie, Prof, and the other Loonies are.
When Mannie asks Prof why he purchased the cannon, Prof tells the story of a man who spent his life shining brass cannons in a courthouse. He saved his money until he could buy a brass cannon of his own and go into business shining his own brass cannon (18.102). The story parallels the story of the Loonies—the courthouse represents the Authority and the Loonies represent the brass cannon guy. Like him, they are still doing what they did before, only now they are it doing for themselves, toward their own happiness and profit.
At the novel's end, Mannie and Wyoh adopt the brass cannon as the symbol on the Luna flag, where it signals a new time for Luna society.