I have a shoebox, wrapped but separate (so looks like a present but I can open it up easily), with a screwdriver, scissors, spare batteries, spare bulbs for tree lights etc - under the tree. Its for the whole season but can be extra useful Christmas morning/afternoon.
I also bring in a black bag for discarded paper and wrappings, and a paper bag for discarded paper that will be reused (to fold that paper into neatly).
I tend to leave things for breakfast organized - like rashers not buried at the bottom of the drawer and the roll of croissants to bake towards the front of the fridge or even in the door. I don't set the table or leave everything out - but I do try and have the makings of a first coffee on the worktop ready to just boil the kettle.
There is also a good amount of fruit and sweets in DD's stocking, so she launches herself into that and is not immediately demanding food from us!
We also get our clothes out for the day the night before - to save rushing too much when going out to mass. DD has a bath on Christmas Eve, and I sometimes have my shower and do my hair then, sometimes on Christmas morning.
We do a lot of the dinner prep the day before, DH and I work around each other in the kitchen and quite enjoy it. peel veg and potatoes, make stuffing, make turkey stock for gravy etc. So we don't have to worry about that before we go out on the morning. We only need to cook when we get in from mass and other visits as we eat in the evening at home - or someone else cooks if we are travelling.
If we are home, I also have a few M&S party food packs - we generally have 1 or maybe 2 in the afternoon - they go in the oven, then turkey goes in the oven, we get other kitchen bits sorted and the fire lit, a drink is poured for everyone, lamps and music get turned on, nibbles are about ready and we take those to the sitting room to relax and open presents, popping back to the kitchen in between times to add other bits to the oven or turn on pots of veg. And I need a good few M&S nibbles the following day for DD's birthday - but there will be lots I can adapt if we are starving on Christmas afternoon and eat more than planned. (It's always savoury ones we use on 25th, a mix of savoury/sweet on 26th).
We have always only had the stocking/Santa in the morning - any presents under the tree get opened once we are home later and it is much more relaxed. Otherwise I would feel like we are rushing too much.
Normally, as the fire can't be lit on Christmas Eve, I clean it out and set it ready to light on 24th, and fill the logs basket etc, so all we need to do is strike a match when we get home on 25th afternoon. (We had it lit last time we were home 2 years ago, so needed to clean it out and set it on 25th - I don't mind so much, but I think I had a "firepac" (bag filled with coal that you just light the bag and have a 4 hour fire before needing more fuel) to just throw in once I'd cleaned it out rather than proper setting with firelighter, sticks, logs, coal.....I tend to keep a couple of firepacs and a good few "firelogs" (a wrapped peat log which gives a 2 hour fire) in the house at that time of year to make it easy on myself).
One other thing - even when we are travelling, and have a rented cottage (necessary for sanity between our 2 childhood homes!), it is just our nuclear family on Christmas morning. We get up, breakfast and head to mass, then go on to the other 2 houses over the course of the day - but don't entertain on Christmas morning. Lots over the rest of the visit - including DD's birthday gathering on 26th - but Christmas morning is just ourselves. (Both our sets of parents are not looking to come - they have a houseful themselves, and want breakfast and mass in either order, and know we'll be over the 1 straight after mass, and the other once the first turkey feast is consumed and cleared up).