Mr. Arrow is one of those little signs we get that all is not right on board the Hispaniola (before, of course, Jim overhears definite proof of the planned mutiny). Mr. Arrow is first mate on the ship, which means he is Captain Smollett's second in command. But he is as different from upstanding, disciplined Captain Smollett as you can get. He's drunk all the time, and no one can figure out where he's getting this constant supply of alcohol. Sure, a lot of the guys in the novel seem to be drunk a lot (since drinking alcohol seems to be one of the key traits of pirate life). Still, it's really serious in the case of Mr. Arrow, leaving him unable to maintain crew discipline.
Mr. Arrow finally disappears, and everyone assumes he fell overboard while drunk. But the two men who fill his duties, Job Anderson and Israel Hands, are later revealed to be pirate leaders. So Mr. Arrow's death leaves room for the pirates to gain even greater control of the ship's management. After overhearing Long John Silver speaking to Dick Johnson and Israel Hands on board the Hispaniola, Jim realizes it was probably the pirates who kept Mr. Arrow supplied with rum from their own stash of booze, leading to his mysterious death.