Goober's Wife

You couldn't actually pay me to set foot in HS on NYE; that park is not built for crowds.  Too many of the walkways especially on the "north" and "west" sides of the park are excruiciatingly narrow.
 
Last year we did NYE in Epcot and it was awesome!  We went in at rope drop and stayed until about 1am.
 
My suggestions to make it a workable day:
1. Wear lots of layers so as it warms up and then gets cold again you can adjust your clothing.  Don't forget scarves and mittens for late night, it got pretty chilly once we sat down on the grass to wait for the fireworks.
2. I don't usually let our kid take toys into the parks but this one day I took his Nintendo DS.  I carried it in my bag and he got to play with it during quiet time.  18 hours of good behavior is a lot to ask of any child and having something to do when we were "waiting" for something helped quite a bit.
3. We took a blanket.  We rented a locker and stashed our coats and blanket in it.  About midafternoon we pulled out the blanket, found a nice patch of shady unoccupied ground, and took a nap in Epcot.  We used the blanket again to rest after 10pm while we waited on the fireworks.
4. We booked two table service meals so we would have 2 chances to go inside and sit down for an hour or so away from the worst of the chaos.  Dinner was in England when they hit midnight in London, so that was bonus fun (and "free" champagne).  Funnily enough, the Showcase walkway outside of Rose & Crown (basically from Canada to the International Gateway bridge) got jammed (completely jammed, nobody could move at all) and the park CMs couldn't get it untangled, so the restaurant staff and CMs escorted us back through the restaurant and out a back way down that little sidewalk that is closed to normal guest traffic. England at 7pm EST is a crazy place...
5. Drink lots. 
 
ETA:  Don't expect to do a lot of rides this day.  Standby lines are VERY long.  We went straight to Soarin at rope drop last year, and I think only rode 2 other rides the rest of the day, maybe 3.  This year since there is now FP+ we will ride those 3 things we reserved, and maybe we'll go to something at rope drop.  But NYE is about the atmosphere and people watching and drinking and eating and dancing, so who wants to stand in line 2 or 3 hours for a ride?
Also ETA:  I would be afraid to try park-hopping on this day.  Park closures are as common on this day as they ever get at WDW.  And you didn't mention how you were planning to move, but transportation is jammed as well.  I would not at ALL recommend trying to move by car after 11am, because the parking lots will be capacity and the roads bonkers.  If you really want to spend the evening in Epcot, it might be worth switching your fast passes and just staying there all day.