“Jack Frost” is the personification of frost (or cold weather). The name was cited in the book Round about our Coal-Fire: or, Christmas Entertainments (1740):
 
“This time of Year being cold and frosty generally speaking, or when Jack-Frost commonly takes us by the Nose, the Diversions are within Doors, either in Exercise or by the Fire-side.”
 
“The Christmas Song” (1945) opens:
 
“Chestnuts roasting on an open fire
Jack Frost nipping at your nose
Yuletide carols being sung by a choir
And folks dressed up like Eskimos.”
 
     
Wikipedia: Jack Frost
Jack Frost is the personification of frost, ice, snow, sleet, and freezing cold weather, a variant of Old Man Winter held responsible for frosty weather, for nipping the nose and toes in such weather, coloring the foliage in autumn, and leaving fernlike patterns on cold windows in winter.
 
Starting in late 19th century literature, more filled-out characterizations of Jack Frost have made him into a sprite-like character. He sometimes appears as a sinister mischief maker or as a hero.
 
Background
Jack Frost is traditionally said to leave the frosty, fern-like patterns on windows on cold winter mornings (window frost or fern frost) and nipping the extremities in cold weather. He is sometimes described or depicted with paint brush and bucket coloring the autumnal foliage red, yellow, brown, and orange.
 
(Oxford English Dictionary)
Jack Frost  n. frost or frosty weather personified.
1826   Sporting Mag. 17 376   Jack Frost, however, put a veto on our morning’s sport.
1872   C. Hardwick Trad. Lancs. 53   The blustering of old Boreas, and the frigid embrace of ‘Jack Frost’.
 
Google Books
Round about our Coal-Fire: or, Christmas Entertainments:
Containing Christmas gambols, tropes, figures, etc. with abundance of fiddle-faddle stuff ; such as stories of fairies, ghosts, hobgoblins, witches, bull-beggars, rawheads and bloody-bones, merry plays, etc. for the diversion of company in a cold winter-evening, besides several curious pieces relating to the history of old Father Christmas ; setting forth what hospitality has been, and what it is now. Very proper to be read in all families.
London: Printed for J. Roberts in Warwick-Lane
1740
Pg. 6:
This time of Year being cold and frosty generally speaking, or when Jack-Frost commonly takes us by the Nose, the Diversions are within Doors, either in Exercise or by the Fire-side.
 
7 December 1765, The St. James’s Chronicle, or the British Evening-Post (London), “Poets Corner,” pg. 4, col. 1:
For the St. James’s’ Chronicle.
To the author of The Summer’s Tale.
THE Criticks, O Cumberland, cavil and rail!
The Reason I’ll tell you thus boldly:
Your Piece should in fact have been call’d Winter’s Tale,
The Audience receiv’d it so coldly!
JACK FROST.
 
5 January 1785, Freeman’s Journal; or, the North-American Intelligencer (Philadelphia, PA), pg. 2, col. 3:
The Life and Adventures of Jack Frost.
And his wholesome Advice to all honest hearts at this nipping season.
A NEW-YEAR’S SONG.
 
Wikipedia: The Christmas Song
“The Christmas Song” (commonly subtitled “Chestnuts Roasting on an Open Fire” or, as it was originally subtitled, “Merry Christmas to You”) is a classic Christmas song written in 1945 by Bob Wells and Mel Tormé.
 
According to Tormé, the song was written during a blistering hot summer. In an effort to “stay cool by thinking cool”, the most-performed (according to BMI) Christmas song was born. “I saw a spiral pad on his (Wells’s) piano with four lines written in pencil”, Tormé recalled. “They started, ‘Chestnuts roasting…, Jack Frost nipping…, Yuletide carols…, Folks dressed up like Eskimos.’ Bob didn’t think he was writing a song lyric. He said he thought if he could immerse himself in winter he could cool off. Forty minutes later that song was written. I wrote all the music and some of the lyrics.”
 
OCLC WorldCat record
Jack Frost
Author: Michael Keaton; Kelly Preston; Mark Addy; Joseph Cross; Troy Miller; All authors
Publisher: Burbank, CA : Warner Home Video, [1999]
Edition/Format:   DVD video : EnglishView all editions and formats
Database: WorldCat
Summary:
On the verge of making it big, long-struggling musician Jack Frost suddenly realizes he’s missing something vital: time with his wife and young son Charlie. But before Jack can make up for lost time, he swerves off an icy mountain road. Then, one cold winter’s night, Jack magically returns as a walking, talking, ski-slope-shredding snowman. Determined to be the father he always wanted to be, Jack embarks on a quest to show Charlie that when it comes to being totally cool, there’s no dad like a snowdad!