Education Innovations sells these neat rings that work like a suction cup to hold your drink.
The original Lil' Suctioner has a radius of just over 2 inches. While I wouldn't go smaller than that, it certainly wouldn't hurt if yours was bigger.
Cut about a one inch hole in the center of your material.
Slip the material over a can or bottle and test it out for yourself.
By the way, the Lil' Suctioner includes some air pressure facts, including the weight of the atmosphere pushing down on it (~221 lbs). But, there's no reason you can't figure it out for your own drink holder. If it's a circle, measure the radius; if it's a rectangle, measure the length of the sides. Make all measurements in inches.
Calculate the area of your material (for a circle: pi x radius x radius; for a rectangle: side x side) and multiply that area by 14.7 psi (pounds per square inch). That will give you the number of pounds of the atmosphere pushing down on your drink holder, of in other words, the pounds of force you'll have to exert to lift up your drink.