"Your father left this in my possession before he died. Use it well."
-- anonymous note explaining the gift
Among the gifts Harry is astonished to receive is an Invisibility Cloak with an anonymous notes informing Harry that it had belonged to his father, and the admonition to “Use it well” (PS12).
When asked whether she had a favourite passage from one of her books, JKR named this chapter, as well as the ending of Goblet of Fire (Sch2).
The writing on the notes is described as "written in narrow, loopy writing [Harry] had never seen before." As we learn much later (PS17), Dumbledore sent the cloak, so this would be his handwriting. Harry recognizes this handwriting immediately in book six when he gets a number of handwritten notes from the headmaster inviting Harry to private lessons.
The note reads, "Your father left this in my possession before he died." The question, of course, is why James Potter would have left his invisibility cloak with Dumbledore rather than keeping it with him. (Of course, if he had retained it, the cloak would have been lost when the house in Godric's Hollow was destroyed, so in terms of story construction that wouldn't work. JKR had to establish right away that the cloak hadn't been in the house on the fatal night.) In terms of the story, consider the circumstances at the time the Fidelius Charm was to be cast. The Potters were going into hiding. Harry was at the top of Voldemort's hit list, something Dumbledore surely told James and Lily. Knowing that, neither James nor Lily would have been going off on any assignments for the Order of the Phoenix that would have required the use of the cloak. My opinion (unverified as yet by any statement in canon) is that James lent the cloak to Dumbledore so that other members of the Order could put it to good use while the Potters were concerned with protecting their son.
The brief note ends with the words "Use it well." If we don't count the Dursleys' "gift", Harry's Christmas presents consist of various things to eat, a Weasley jumper, Hagrid's flute, and the invisibility cloak. The flute later gets Harry past Fluffy, and the cloak has been an immensely useful tool on many occasions. It's worth comparing this to the arrival of Father Christmas in C.S. Lewis' The Lion, the Witch, and the Wardrobe, chapter 10 "The Spell Begins to Break" (the White Witch's spell of winter had meant that it was always winter and never Christmas in Narnia). When Father Christmas presents his gifts to Peter, Susan, and Lucy, he begins by saying "These are your presents... and they are tools, not toys. The time to use them is perhaps near at hand. Bear them well." (Emphasis mine.) And after presenting the tangible gifts, Father Christmas made sure to conjure up a little Christmas tea, so food and drink weren't forgotten there, either.
Harry was perfectly prepared to have the best Christmas of his life to date without expecting any presents at all. The food and the fun would have been enough for him.