How to add weights to circuit board standoffs(spacers) of a acrylic board?
For your purposes, one can expect that more weight, as low as possible, will accomplish your objective. With circuit board standoffs, you have a threaded hole at one end and a threaded stud at the other. This provides for easily attached weights in various forms.

You've considered coins, which if chosen successfully, can be attractive. Think various sizes of coins from countries around the world, with holes drilled to accept the bolt you'll use to attach to the standoff. 

US coins are somewhat lightweight and poorly suited for such a project, in my opinion. Masses composed of lead will provide the best weight per volume but lead has its own complications. It's not dangerous to handle but there are some health considerations. I have a cupcake tin that I use to make lead ingots, about 80 mm in diameter and 30 mm thick, probably too large for your purpose. I then dipped them in a product called PlastiDip which is also available in spray form. 

If this is attractive to you, it's a matter of picking a suitable mold (silicone molds can work) and finding someone to pour. Fishermen will frequently create their own sinkers and avid weapons owners will cast their own rounds, providing at least two resources for you.

If lead doesn't cut it, steel would be the next available option. Steel washers, both stainless and galvanized forms are available at big box stores and mom-and-pop hardware stores as well. Pick a size for your taste and aim for either the hole size to match the thread size of the hole, or match the hole size to the outside diameter of your stand-off.

Even with the possibly-unsightly washer option, you can still apply PlastiDip spray to provide a colorful rubber coating for surface protection and cosmetic appeal.