In the manner of most cats, the name that sticks was acquired almost by stealth, and can be taken as shorthand for "Stop that, You," which is a milder version of the name "gerrofoutofityoubugger!" which is the pronunciation of Greebo used by most people who have met him.
Tiffany is in some obscure way testing out Granny, a known ailurophobe, with her gift of a cat, viewed in many cultures as the essential accessory for the working witch, or else a natural development in the life of an eccentric old spinster who lives on her own.
Granny has reluctantly accepted the burden of being owned by a cat, (naïve people refer to this as "cat ownership", which is correct, but then get the relationship of "owned" and "owner" exactly wrong.)
Thus far, You has trained Granny into taking her everywhere she goes. As a kitten, You reclined in the warmth and comfort of the space between Granny's tightly bunned hair and pointy hat. As of I Shall Wear Midnight, she drapes herself across Granny's shoulders like a recumbent queen, from which venue she might (rarely) grace someone with the immeasurable honor of blinking at them for a second or two, before losing interest.
You's self-evident mastery of headology has ensured that she eats better than anyone else in the Weatherwax household: whilst Granny and Miss Tick dine on watery bacon soup, You enjoys the luxury of chicken scraps.
There is a fitting symmetry to one of the most powerful Witches on the disc, a woman dressed in the traditional black, being owned by an all-white cat: this may partly be due to Tiffany's known dislike of the colour black and partly because all-white, the colour of the full moon, is in itself of significance in the arcane world of Witches magic.
She reappears, fully-grown, in I Shall Wear Midnight as a stately and queenly cat, borne in state across Granny's shoulders.