A giveaway box is a container full of interesting items that can be handed off whenever needed. You want a reasonably sturdy box of good size. One used for holding reams of paper is nice and sturdy, with handle holes and a lid, but you can use whatever you have. Keep it somewhere out of the way but convenient. Just off the living room makes it accessible if you often use it in public, but the back of the house is more discreet if you tend to use it secretly. Do what works for you.
First, you seed the giveaway box with cool things. They can be used in decent shape, or new, it doesn't matter which. It can have a theme or not; we have one just for books that we call the hobbit housewarming box after its first appearance, and one for miscellaneous items. I refill the miscellaneous one by watching for nifty things on sale cheap. Good costume jewelry and crystals are a favorite with Pagans, pretty little boxes or pouches work for anyone, tiny games or dice/cards with interesting designs are also nice, cute fidget toys, etc. Don't overlook practical things like pencils or rulers that people are always running out of -- I have occasionally bundled spare pencils or pens into the box, and had people squee over them.
Also if you live in a gift economy, even a subculture one, after folks know you keep a giveaway box, they'll often bring you handfuls of things for it. "I was decluttering today and don't need these anymore." I've had friends hand me whole Tarot decks, books, and stone figurines that way.
You can then use the giveaway box in at least two ways:
1) Someone shows up that you don't have a gift for and needs one. Slip into the back of the house, grab something suitable from the giveaway box, and wrap it or pop it into a gift bag. (Keep those supplies nearby.) Then present the gift to your guest.
2) Pass it around a group of people with the instruction, "Put one thing into the box and take one thing out." This is an excellent option for people who dislike conventional gift exchanges because they have trouble choosing gifts for others or getting anything they like. It's free choice. If you're setting up this activity, it helps to give a price guide for buying new, but often the best stuff is handmade or used.
Basically, this is a modern version of the giveaway which has many forms throughout Native American culture. I got it from my grandmother, and it probably came from a Cherokee ancestor. It's fairly widespread across much of the South, and yes, it likely spread from various tribes. You also see it in Pagan circles, and among fannish folks people may call it a hobbit box or a mathom box. (Mathom is a hobbit word for things of value but no immediate use, which are often passed around as gifts.) I'm glad I could share the custom with somebody new. Have fun with it! :D