For sure, being a holiday week, it's going to be busy and expensive. With New Year's falling on a Wednesday this year, I would suspect that the latter part of that week and the Jan. 4-5 weekend will be better than the previous for finding something reasonable. GIven that you have a non-skier, I do highly recommend that you find a place in Park City as opposed to outlying areas. Cheaper options would be at the I-80 intersection around Kimball Junction, Newpark, Redstone, etc. There's a Holiday Inn Express, Hampton Inn and Best Western. Also check Newpark Resort and maybe even VRBO.com for condos in that area.
Conditions should be similar to what you find in Colorado that time of year, but do consider that elevation is about 1,000 feet or more lower than comparable resorts there.
For the drive over, I-25 to US287 in Fort Collins to I-80 in Laramie is fastest way over. Do watch weather in Wyoming as it gets nasty there on occasion.
Where does your son race out of?