We played that last year and the person unwrapping had to wear oven mitts. We just did lottery tickets and candy. If you do paper money, I would suggest folding it up and putting it into one of those little cardboard gift card holders so it doesn't get ripped. Maybe it's just us, but we are rather spirited with tearing the Saran wrap off and dollars would have likely ripped.
Some other ideas you could do are coins (half dollars, gold dollars, foreign coins if you can find them), small keychains, lip balm, Paracord bracelets, the little surprise pack toys, glow bracelets (if you make it shortly before the game- go ahead and snap them as you wrap them and the ball will glow), winter gloves, the dollar store expandable washcloths, liquor minis, fruit snacks, mini slim Jims, hair ties/headbands/hairclips/mini compact hairbrush, mini flashlight, pocket/keychain tools, mini stuffed animals (Walgreens has cute mini dogs with a Santa hat that make noise), squishees, thumb drives, power banks (take out of box and give the box to recipient after), led ice cubes, Mardi gras-style beads, lighted Christmas necklace, earrings, neat lanyard, Disney pins, face mask,
Lego minifigs, finger lights, Christmas socks, gum, deck of cards, hanging purse clip, ResqMe tool, cool magnets, red Rudolph nose (thinking the red nose day noses), earbuds, small ornaments that won't break, cookie cutters, finger puppets, small bottle opener, boo boo bunnies, hand warmers, sleep mask, folded up reusable bags, neat soap, mini Play-Doh or slime, small bottle of Poo Pourri.
If they don't receive something age or gender appropriate, it just makes it funnier and they can always swap at the end. Have fun!