My brother recently got a 3D printer which caused me to get online and see what sort of things he might be able to manufacture. My search resulted in a few interesting business card related gadgets.
How about a braille business card writer? This simple device will allow you to easily emboss any business card sized piece of paper with your company contact details in braille, making you instantly ADA compliant, and delighting your visually impaired friends and business associates. As a former president of the Chicago Uptown Lions Club I found this device particularly interesting. It is probably one of the more complicated devices I found but also one of the most innovative. Likely not a good choice for your first project however you might want to check this out then “press” it into service.
https://pinshape.com/items/34568-3d-printed-braille-business-card-writer
I was able to find plans for a few different business card holders. This first one is basically a small business card sized box with an opening on one side for easily getting ahold of the cards. Kind of like a little recycle bin. It also has a small channel on the rim where you can slide in your card for display.
https://pinshape.com/items/64798-3d-printed-business-card-holder
The next desk accessory was an attractive negative space cup-like design for holding pens, pencils, letter openers, plastic sporks, an extra pair of reading glasses or whatever. The base has molded slots specifically designed for holding various digital media storage devices like portable USB flash drives and also SD cards. On the side is a rectangular space for business cards.
https://pinshape.com/items/38602-3d-printed-businesscardpenusbsdmicrosd-holder
The last of my list is the most simple design, utilizing the least amount of plastic material. Basically a kind of wire-frame that allows you to display about 20 cards upright in a traditional manner where they can be easily viewed and offered. The plans include a vertical as well as horizontal display option.
https://pinshape.com/items/32686-3d-printed-business-card-holders
Look around a let me know if you find anything else of if you try printing one of these options.