I don't do Xmas, but I do do food gifts. I tend to save jars, jam, sauce, whatever. I have also bought small jars from the pound shop.
Things I have made
Pickled onions, but with a small amount of raspberry vinegar added to the spiced vinegar with extra pickling spice.
Things preserved in oil - I buy pomace oil but if you have more budget a better olive oil.
Get some Greek style supermarket cheese, cheaper than feta but still works and a packet / jar of dried mixed herbs.
Sprinkle the herbs on a worktop / chopping board and press each surface of the cheese then cut in to chunks and again press every surface in to the herbs.
Put the herby cheese in a jar and cover with oil.
I also do a mix of finely sliced olives, peppers, mushrooms that I put in a jar with a clove of garlic and some capers (you could add anchovies but I am not a fan) cover with oil and leave for a week or two to infuse.
Chutneys, I'm in Wolverhampton so lots of Indian shops selling mangoes and other fruit / veg that costs a fortune in the supermarket.
I have also made spice mixes and add a recipe for the curry it will make. This goes down well with people who do not have the same access to diverse shops (my brother lives in Cornwall).
It's not just food that can go in jars, you can get cheap battery powered LED lights that make a nice decoration.
I actually have quite a few LED tealights that I used yesterday to decorate the path outside. They can go in a jar with a bit of holly and ivy (both grow in my garden)