Embossed business cards are among the most interesting
If you're not familiar with embossed business cards, basically what happens is a die (and counter die) are used to push up the paper from the back. In other words, one side of the die has the image raised, the other side of the die has the image indented... so when the paper goes between it, it molds into the shape.
One of the benefits of an embossed business card is it's ability to stand out. It looks different. It feels different. And it elicits that 'hey cool!' reaction that every business owner hopes for. After all, the more of an impression your business card makes (and the bigger the conversation it creates) the more likely it is to achieve its desired result. That is, generating more business.
What to emboss?
Well, you can emboss your logo or any other shape you can think of. Like a dentist for example, who might emboss teeth marks to suggest someone bit down on the card. You're limited only by your creativity.
Embossed business cards even have some customization options. For example, you can blind emboss, whereby the image is simply raised. There is embossing with ink which coats the raised area with color. And there is foil embossing as well, which coats the image with a shiny metallic coat.
There are a couple of drawbacks to embossed business cards however.
First is the additional expense. Due to the custom design, die creation, and effort required there is often considerable additional expense compared to some other stand out options. Naturally, they can't be produced as quickly either. For many, this may not be a big deal as the expense can be outweighed by the added benefit and response your card generates.
Second, printing on the back of embossed business cards is not advisable (or more difficult) for obvious reasons. Proponents of double sided cards may be deterred by this.
Finally, (and similarly) an embossed business card generally has less flexibility and less room for content, limiting the amount information on your card to just the basic contact information. For some this ok, but those who want to add additional content, selling points, bullets, offers, images etc. in addition to basic contact information, will find that an embossed business card may not be the best option.
What your decision should come down to is this:
Is the idea you have for your embossed business card (as in the shape of the die you intend to create) creative, unique, and attention grabbing enough to justify it as an option? Is embossing your logo, for example, going to elicit the reaction you're looking for?
Or are there other options that allow you to be unique -- like 3d lenticular cards (flips back and forth between two images), plastic business cards, die cut cards, or fold over cards, etc. -- that offer you greater flexibility and opportunity to stand out, without limiting your card's content?
As long as you're thinking about the ultimate goal of generating clients, your decision... will be the correct one.