Through out the year I pick up things that I know someone in the family will like or can use. If it's small or generic things, then I can usually still afford to get them something they specifically ask for (if they do - my family rarely does).
This year I'm giving mostly hand made gifts - depending on what you make & where you buy your materials, you can save a lot of money & not have to worry about crazy shopping days. Assortments of cookies & bars make good gifts, as do bread or muffin baskets w/ a homemade spread. Attach the recipes & use some festive wrapping & you're good to go

I also like making soap sets & this year I'm making scrapbooks for a couple people.
I also like to take advantage of the leftover holiday goods - you don't have to go shopping during the crazy sales, but there's always leftover wrapping paper, bows, bags, etc. marked wayy down after christmas.
Speaking of markdowns - I love giving books as gifts as well, & there are lots of cheap places to buy them either new or like new.
& I try to ask people early on what they want, so I don't have to try to find whatever it is they asked for in the middle of december.