Having a card as a way of providing information to someone is one thing, creating a card that stands out is another. There are many ways to create a unique business card, but most of these methods could cost more than "factory" produced business cards, but the expense might not be exactly huge compared to your overall marketing budget.
Add rounded corners or a combination of rounded and rectangular corners.
Use engraving to create a superior masterpiece in which the text appears above the rest of the card. You will find several engraving printers available in large cities.
If etching is a lot for your finances, consider thermography, a process that emulates etching, with corresponding effects, though not as satisfying for the purist. You may get more information about metal business cards via https://www.puremetalcards.com/.
Foil stamping, perhaps over part of your logo, creates a card that stands out from others.
Most color cards are produced using offset lithography or digital presses. Consider producing your card using traditional letterpress printing. The result will likely have a slightly more three-dimensional feel.
Create a bump effect or a bump effect. Effects can be produced without ink, i. me. 'blind' or inked.
Don't forget the back of the business card. This is a good way to provide more information about your business, for example, you could include a map of your physical location or list your social media accounts.
Consider creating several different business cards for yourself, perhaps with the same but slightly different themes.
Note that the cards do not need to be printed horizontally, they can even be printed vertically.