What happens when a creature with Persist AND Undying dies? ('return to battlefield' issues...)
Both abilities trigger, and you decide the order that they resolve. If you choose to have Persist resolve first, then when Undying is trying to resolve, it fails to find the creature in the zone that it expects. Undying could normally find the creature that moved from the battlefield to a public zone (the graveyard), bu when Undying triggered it was looking for the Object that got moved to the graveyard where the object no longer is.


  400.7. An object that moves from one zone to another becomes a new object with no memory of, or relation to, its previous existence. There are seven exceptions to this rule:
  
  400.7d Abilities that trigger when an object moves from one zone to another (for example, "When Rancor is put into a graveyard from the battlefield") can find the new object that it became in the zone it moved to when the ability triggered, if that zone is a public zone.


This all, is of course assuming that both abilities triggered (you said that they did). It is even possible for both abilities to not trigger (as ikegame mentions). This could happen for example if your Undying+Persist creature was dealt damage by a source with Wither and it had a +1/+1 counter. State-Based Actions would be checked, and the creature would be sent to the graveyard at the same time that the counters got removed.


  704.5f If a creature has toughness 0 or less, it's put into its owner's graveyard. Regeneration can't replace this event.
  
  704.5r If a permanent has both a +1/+1 counter and a -1/-1 counter on it, N +1/+1 and N -1/-1 counters are removed from it, where N is the smaller of the number of +1/+1 and -1/-1 counters on it.


The Comprehensive Rules give an example for Undying illustrating this:


  704.7. If a state-based action results in a permanent leaving the battlefield at the same time other state-based actions were performed, that permanent’s last known information is derived from the game state before any of those state-based actions were performed. 
  
  Example: You control Young Wolf, a 1/1 creature with undying, and it has a +1/+1 counter on it. A spell puts three -1/-1 counters on Young Wolf. Before state-based actions are performed, Young Wolf has one +1/+1 counter and three -1/-1 counters on it. After state-based actions are performed, Young Wolf is in the graveyard. When it was last on the battlefield, it had a +1/+1 counter on it, so undying will not trigger.