So my mum started making reusable fabric bags....oh probably 10-15 years ago with left over scrap fabric pieces and it became a running joke in our family that if Mary gave you a gift she wanted the bag back.
We still have some of the very first fabric bags she ever made.
They can be made easily with a sewing machine, scrap fabric (old tshirts would be great!) and some string or yarn to tie them off. You can make them occasion centric (Christmas, Birthdays, Baby Showers, Weddings, etc) and they can be made in any size!
And you can be like my mum and ask for the bags back OR if you're stuck for gifts make a bag of bags and spread the sustainable wealth!
It's really a perfect project for beginner sewers.
This site reuseit.com has options if you aren't savvy in sewing at all! But there are other sites as well.
Someone recently told me that wrapping things in bags is "cheating" and "takes the fun out of it" but honestly, fun to me is being able to see the planet healthy and alive.
If you do use wrapping paper be aware of recycling standards in your area and take the time to separate and do your part.