Adding a little flair to the top of your leftover containers is the easiest way to make your leftover gifts look impressive without using a lot of materials. All that's needed is some holiday-themed or colored paper, tape, and an Exacto knife. Press the piece of paper into the ridges of the lid until they're well-defined, and then use the knife to cut out the circle before affixing the paper to the lid with tape.
This technique is especially perfect for mason jars. After adding the paper flair to the lid, the ring will affix it to the jar, and give it a charming, framed lid. This holiday look will transform your mason jar into the perfect holiday vessel without removing the charm of the glass itself.
Alas, not every vessel is as attractive as a mason jar. Old plastic containers (think yogurt or sour cream) can undergo an easy transformation by treating them like any gift. Grab some wrapping paper, tissue paper, or even plain brown butcher paper and wrap the cup part of the old container as if you were wrapping a gift, and secure with scotch tape. Tissue paper is especially forgiving when wrapping irregularly shaped cups, because you can make creases along the height of cup so that the paper hugs the cup. When you add the papered lid, the container becomes a usable and charming, fully-wrapped gift. Likewise, be on the lookout for holiday-themed food boxes. They're a cute way to add a little extra pop to your hand-made gifts.
Once the food is wrapped, labels will make it clear to your guests what's what, and ensure that you don't mistakenly give them five jars of potatoes and no turkey. A simple brown tag and string can do the trick (you can make them out of manila folders, and amp them up with a little coffee dying), you can use a plain white sticker label, or print out any of the many downloadable tags available on the Internet, like Disney's Family Fun Thanksgiving labels (used in the pictures in this post). This is also a great way to offer re-heating instructions to your guests who are hopeless in the kitchen.
Your guests will need something to carry their tasty gifts home, and plain, brown paper bags with handles are the perfect solution. They're cheap (you might even have some saved from recent shopping trips) and when they're teamed with some orange or brown tissue paper, they will become appropriately seasonal in a snap, and are also the perfect canvas for stickers and holiday ink stamps.
Use one of your labels to offer one last bit of thanks, and to let each guest know which bag is theirs. (This will also come in particularly handy if allergies or food restrictions are an issue -- everyone will know what goodies were packaged specifically for them.)
Other possibilities include: Reusable canvas bags, traditional gift bags, grocery store brown paper bags, or even a wrapped shoebox.
The above basics will result in some beautiful gifts for your family and friends, but if money and time allow, you can always take it to greater levels. Tie a long cinnamon stir stick and bag of mulling spices to each bag for an aromatic and useful "bow" to the gift. Use Thanksgiving-themed stamps on brown paper to give the gifts a whimsical feel. If you want to give a gift that lasts, package the leftovers in new Tupperware containers that can be washed and reused, and bag the leftovers in festive reusable lunch bags.