‘Office’ meant that Veronica had a feastday with readings and prayers specific for the occasion.
The problem (and the reason why St. Charles Borromeo excluded the office from the Milanese Rite) is, we really do not have any solid historical information about Veronica. Originally, ‘Berenice’ or ‘Veronica’ was a name given to the woman who was healed of her hemorrhage when she touched Jesus’ clothes. (Like how ‘Dismas’ among other names was the name given to the good thief.) So the only thing we know for sure about ‘Berenice/Veronica’ other than what the gospels tell us: she had a hemorrhage for twelve years and was cured when she touched “the hem of [Jesus’] garment.”
Now as time went on, in the West a story about a miraculous image of Jesus got attached to this woman. Originally it was a
man-made image, a portrait of Jesus or on a panel she had commissioned so that she would have something to remember Jesus by. (A much earlier story had the cured woman build a
statue of Jesus curing her as a tribute!)
But the story grew more elaborate with each retelling: soon it was now reputed to be an image ‘not made with (human) hands’. In this version, Jesus rendered the painter (in some later versions of the story, this mystery painter is claimed to be St. Luke) unnecessary by going to Veronica’s house Himself and wiping His face with the canvas the painter intended to use, miraculously imprinting it with His image. (It was pretty much the Western version of the Eastern story of
King Abgar and the Holy Mandylion, a story which went through much of the same development.) By the late Middle Ages, the story evolved again: whereas the earlier versions of the Veronica story place the event somewhere in Jesus’ ministry, now Veronica is imagined to meet Jesus as He was carrying His cross to Calvary; the painter’s canvas or panel now becomes her veil or handkerchief that she wipes Jesus’ face on as an act of compassion.
What really complicates things is that you have this relic venerated in an oratory in St. Peter’s Basilica since around the 11th century, confusingly
also called ‘veronica’ - in this case, a corruption of the Latin-Greek phrase
vera icona ‘true image’ - and often also reputed to be a miraculously-formed image. One has to distinguish between ‘Veronica’ (the woman) and ‘the
veronica’ (the image/relic), but the annoying thing here is, folks back then
did confuse and interchange one with another: the portrait Veronica possessed was identified with the veronica in St. Peter’s. (Though note that our earliest testimony to the existence of the veronica in St. Peter’s (from 1160) gives a different version of the origin story of the relic: there it is claimed to be a towel Jesus wiped His face on when He sweated blood in the garden of Gethsemane (no woman is involved here, nor is an image explicitly mentioned).)