According to Niall Ferguson in The Ascent of Money, they aren't balls, but coins. (I listened to the audiobook so I can't provide a page citation.) I'm somewhat suspicious because the blazon for the arms is
"augmented coat of arms of the Medici, Or, five balls in orle gules, in chief a larger one of the arms of France (viz. Azure, three fleurs-de-lis or" Source. That source seems to have problems, so check wikipedia sidebar
Note that the first five or six google references refer to them as "balls"; although I'm not familiar with it, "balls" is apparently a legitimate synonym for "roundel", and perpetrate the notion that it is a mystery. None of these are particularly scholarly.
The Rise and Fall of the Medici positively asserts that there is no relationship between the Medici arms and the symbol for pawnbrokers. Another myth is that they are pills, as a reference to the name of the family "Medici = Doctors". While "canting arms" are common, this doesn't explain why they are red. Furthermore several sources suggest that
(the story that these were medicinal pills was apparently invented at the court of France by people wanting to defame Queen Caterina dei Medici in the 16th century) Florence Art Guide and several other sources (hat tip to @Yannis Rizos who first pointed this out).
If they were intended to represent coins, the more logical charge would have been the bezant, which is a gold Roundel Of course the field is or, so the charge must be a color; making them red makes them torteu or "cakes". However, I think that is a clue.
The answer which seems most likely is in a footnote; the source it cites is behind a paywall, but this looks quite plausible.
According to one theory, the armorial bearings of the Medici are canting arms or armes parlantes, and the torteaux or red balls supposedly represent pills, because medici in Italian means "physicians." The historian G. F. Young regards this whole story as a fable.�The Medici, chap, iii, n. 2. He is probably right. A more plausible explanation is that the Medici adopted the roundels because they were the symbol of the banker's trade and of the guild to which they belonged. The coat of arms of the Florentine money-changers* guild, Arte del Cambio, was a red shield sown with bezants or gold roundels. The Medici used red roundels instead of gold ones. The pawnbrokers eventually adopted the gold roundels or balls as the sign of their trade, since those symbols were associated in the public mind with money lending and credit.�Raymond de Roover, 'The Three Golden Balls of the Pawnbrokers," Bulletin of the "Business Historical Society, XX (1946),
117-24. See esp. illustrations on p. 123.
Although the name means doctor, the family Medici rose to prominence due to their banking, and they are very tightly associated with banking. I think it is very plausible that their coat of arms is a reference to the money changer's guild.