Popecoin:  a million new mining blocks released every new pontiff's election
Saintcoin:  a million new blocks released every time a new saint is canonized (bonus:  each saintcoin is automatically a fourth-class relic of that saint!)
Denominacoin:  a new block released every time a new Protestant denomination forms (approximately every two minutes or so?)

I'm sure Buddhist, Zoroastrian, Taoist, and even Communist versions could be possible.