According to the Christian legend, and to quote Wikipedia, Saint Veronica (or Berenice) was a pious woman of Jerusalem who, moved with pity as Jesus carried his cross to Golgotha, gave him her veil that he might wipe his forehead. Jesus accepted the offering and after using it handed it back to her, the image of his face miraculously impressed upon it. The name of the holy woman is probably an alteration of vera icona, the true image.
As you might imagine that kind of story has left a big impression on painters, and it’s not a surprise if the images of Veronica and her veil are so numerous. How painting could be unchristian if God himself dabbled once in that art? Moreover, it was an occasion to demonstrate virtuosity in executing the painting of a painting.