This is probably obvious, but it'll be colder the further north you get - that's true year-round. When I was first in Italy in August 2001, it was stifling in Rome and downright chilly up where our friends live (base of the Dolomites, an hour north of Venice). Remember also that Turin was the home of the last winter Olympics, so the real north of Italy is a snow bunny's playground in winter.
That said, I imagine Rome would be pretty cool just before Christmas (I'd second a recommendation to see Vatican City - I'd also recommend going with a guide; we did a half-day tour of the museums and St. Peter's and it was well worth it).
Overall, without knowing exactly what cities your friend wants to see, I'd say wintertime (or any time with colder weather) is a good time to visit any place where the main thing to do doesn't necessarily involve being outside. In Italy, you've got more museums than you could visit in a lifetime - and when it's cold outside, it's a perfect excuse to spend time in a nice, warm museum getting cultured. If your friend has her heart set on Venice, I'd just be a little careful about that one - I adore Venice, don't get me wrong, it's just that in the winter it can get flooded. It's supposed to be amazing to see, but more difficult to get around. And that wonderfully romantic gondola ride ain't all that when you're freezing.
And of
course I want an excuse to go back to Italy...