It will be helpful if you can clarify a few things.
First, are you interested in the equilibirium concentration of the solute or the rate of dissolution?
Second, are you dealing with high concentrations or low concentrations?
Third, are you dealing with volatile solvents or non-volatile solvents?
Fourth, what temperature range are you dealing with?
Fifth, are there any special materials considerations? For example, are the solutes or solvents corrosive or have other properties that require special considerations?
Sixth, when you say "ionic liquids" are you referring to something such as a molten salt, or perhaps a weak organic acid that is only weakly ionized in pure form, or some other type of material that one might call an "ionic liquid"?
I think if you can clarify questions such as these it will help others formulate good answers.
your question is very much justified. Solubility measurement is always tricky. If approach to equilibrium is very slow (case of low solubility) too low values maybe obtained. However, it is difficult to avoid and distinguish dissolved and dispersed solutes. This is difficult to be avoided for the ionic liquids because of there amphiphilic nature.
For very precise measurements layering of ionic liquid and solvent maybe necessary followed by analysis of ionic liquid in solvent layer as function of time.
As a first approach mixing of both followed by centrifugation may work.