While it wasn't at Xmas, I was there in early Jan. two years ago and while it was cool (around 6 Celcius daytime high) there wasn't much rain, although I think I may have lucked out as the locals kept reminding me to take my umbrella. I wouldn't expect monsoon season, but steady drizzles.
A good, lined, waterproof windbreaker or jacket should do the trick along with study shoes to puddle jump and/or stand with the crowds at Piazza San Pietro. I forgot to pack gloves, an unpardonable sins for a Canadian I know
, but something light for the hands will keep the cool off the early morning walks. Lastly, I'd recommend a scarf that not only will keep the cool from around you neck, it just might have the added bonus of making you local like a local.