Your common or garden philosopher will refer to the Ship of Theseus, or Theseus paradox, and those with a foot in the '90s will call it Trigger's Broom.

Those truly tuned in to popular culture will indeed refer to The Sugarbabes, a UK Girl Group who presented a similar situation when all their original members were replaced one by one. This is the superior reference because eventually all the original members returned, elevating the situation to new levels of paradox.