You need (1) Any solid in powdered form (2) Recommended: Any other product that would change the nature of the solid to make it soluble in the solvent (let's inaccurately call it "solubility enhancer"), (3) Any solvent in which this solid would dissolve without producing gases or unsoluble co-products.
You would pick the solid and solvent based on:
A) How much solid you would like to dissolve in the liquid. The Ksp (Solubility Product Constant) would be a good indicator of this, telling you how many molecules you can dissolve in a given volume of solvent at the given temperature you'd like to do your demo. If you want to have a lot of sediment that would disappear completely, you would need a compound that is highly soluble in that solvent and at that temperature (in the form resulting from the reaction with the solubility enhancer if you chose to use one).
B) How quickly you want it to dissolve (i.e. "dissapear"). You can influence on this independently of the Ksp in at least to ways: The first is how finely ground your solid is (as it influences its contact surface), you could go from pellets down to "nano-powders", the second is how the reaction itself will influence the conditions determining the speed of the reaction: the solubility of many compounds increases with temperature for instance, and some reactions give out energy (called "exothermic" reactions), which means that you could chose a reaction that increases it's own rate of reaction as it proceeds (carefully within the limits necessary not to increase the container's temperature or internal pressure beyond what would be safe for your demo).
I would recommend one setup whereby:
(1) The solid to be dissolved is first set at the bottom of your container
(2) The solid to be dissolved is covered by a very thin layer of another solid densely packed in such a way as to insulate the first layer as much as possible from the solvent. This solid won't dissolve but the relative quantity is so small that no one will notice.
(3) The solvent is then very carefully added as not to mess up the two layers of powder
(4) You keep the recipient untouched, preferably right were you need it for the demo
(5) Right before your demo you carefully add to the solvent the chosen "solubility enhancer"
As you start shaking the bottle, the insulating solid layer will disperse, allowing for the underlying solid powder/pellet? mass to contact (1) the "solubility enhancer" it will react with to create a more soluble product (2) the solvent it will dissolve in. Now based on the solvent, the solid, and the product you make it react with, you could be shaking for a dozen seconds, or for a couple hours. :)