Can I just point out that the aims of Buckingham's Rebellion were to A) put Edward V back on the throne, and when they heard that he might be dead, became B) to put Henry Tudor on the throne? Buckingham had no intention of leading a new faction, he just opposed Richard III and intended to replace him with a better candidate.
The real issue here is that three-way wars simply don't happen. You'll always find it hard to add a third faction to anything, because there's usually no convincing reason why the third faction wouldn't in fact be part of one of the existing two sides.
You are correct. However, it took a very specific set of circumstances for him tolaunch the rebellion of OTL. Without the two Ricardian coups of 1483, Tudor is still a nonentity, and Buckingham a loyal Yorkist. My reasoning is that he's one of the few people remaining by the time Edward IV kicks the bucket. Hastings is unlikely to betray Edward and his sons, whereas Buckingham obviously did so in OTL. However, for Stafford to set himself up as a third party in the struggle, we probably require a much more turbulent period between 1475 and 1483. Of course, if that's the case, there's no guarantee Buckingham survives this period.
It's not very likely that anybody can do it, but of those very long shots, Buckingham is probably a nose in front of the others. The only other candidate that springs to mind is John de Vere, Earl of Oxford, or his father or brother if they survive. Yet again though, the writing would need a reason for him to go it alone, instead of supporting Tudor as in OTL.
Best PoD for such a thing? I would have Edward of Westminster survive Tewkesbury and escape back to the continent. This sets up a more turbulent 1470s. Added bonus if there's serious domestic turmoil after the Treaty of Picquigny.
Of course, the result of this extra bloodshed might just be that an obscure outsider comes to the head of one of the current factions (see Tudor, Henry for details), rather than a third side appearing.