It's Christmas Eve-Eve and if you're like me, you are about to start wrapping your presents for your entire family before you leave tomorrow night for a six hour drive with your mother and tiny old dog leaving you no time once you're in New York to wrap them before Christmas morning. (OK, you may not be doing it exactly like that, but it's close.) I haven't got a ton of extra fabric that I need to part with, so furoshiki! (the exclamation is necessary) is not going to be my method of choice for last minute wrapping, but it is hella-cool. I imagine you could use some fabric presents as the gift wrap for others, too -- is anyone on your list getting placemats or towels?
Furoshiki Gift Wrapping - Funny videos are here
A pretty basket or box could be the gift wrap, too; everyone needs more storage solutions, and these can be found pretty cheaply new, or better yet, for dirt cheap at a thrift store.
If you are just looking for alternative paper, a newspaper with some ribbon makes quite a pretty picture (and if you don't believe me, I've seen similar at Anthropologie, and they know set-decorating like nobody's business). I'd recommend the classified ads, as all text with minimal disruption reads more like a neutral than headlines covering your presents. And colorful ads from magazines are good for small presents (tape multiple pages together for larger items).
This year, I'm using site plans from my office's recycling bins. I got ambitious last week and colored the fronts of buildings in the site plans with green garlands with red ribbons across the windows, but the black on white looks nice too. Last year, bus timetables taped together worked for me. (WMATA prints them in alternating hues of green, so it fit right in with my extended family's more traditional wrapping paper.) And don't forget just plain reuse of regular wrapping paper. We would use minimal tape and thriftily flatten the paper each year when I was younger to rewrap presents the next year, and it was not a big deal.
Hat tip to 365daysoftrash for the furoshiki! video. Quite a neat and thought-provoking blog he's got there.