How to predict the solubility of an organic compound in different kinds of solvents?
There are some general guesses one can make from looking at the structure but the Abraham solvation equation is commonly used to estimate the solubility of a compound in a given organic solvent:
$$\log P_s = c + e E + s S + a A + b B + v V$$
This equation relies on a set of descriptors to characterize the properties of solvents and solutes to give the water/solvent partition coefficient $P_s$ (the ratio of the solubilities in water and solvent, under certain assumptions):

$p,e,s,a,b,v$ are coefficients that describe the solvent. Abraham and coworkers have collected coefficients for 85 different solvents.

The other variables are descriptors of the solute:

$E$ is the excess molar refractivity, which is a measure of the polarizability of a molecule.

$S$ is the solute dipolarity/polarizability.

$A$ is the overall (summation) hydrogen bond acidity.

$B$ is the overall (summation) hydrogen bond basicity.

$V$ is the McGowan characteristic volume.

You can look at the paper if you want to know exactly what these descriptors are, but their values can be obtained through various experimental and computational means, and combined with the solubility in water, used to estimate the solubility of a given compound in an organic solvent.

Fortunately, you don't have to do all this by hand as several websites, like this, that offer lookup and computation where you only have to specify a compound and pick a solvent from a list.