How to separate sodium iodide dissolved in water?
I summarize some answers given in comments.

Evaporation is really one of the standard methods in any combination of solid (cristalline, amorph, polar, unpolar) and solvent (polar, unpolar, flammable...).
But you must not think of it as the pot heated on a flame. 
In modern synthesis you often already work with a nitrogen or noble gas atmosphere in a closed apparature and use varying pressure for controlled destillation. 
So it is no problem to just reduce the pressure and evaporate your (organic) solvent. 
In your special case you even can think of a pot heated on a flame, there is no risk and disadvantage. It is simply an effective method.

Do you know the concept of polarity? A good rule of thumb is that polar substances mix with polar ones but not with unpolar ones.
So water dissolves salt but not paraffin wax. 
Benzene on the other side dissolves parrafin wax but no salt and water and benzene do not mix with themselves.

=================Edit (Addition of Information, because of question in comment)

So let's see how to regain the solvent. 
First of all there is not only one boiling temperature. The boiling temperature is always a function of pressure. Approximately it can be described by the Clausius-Clapeyron-Equation.
$$
T_2(p_2)=(\ln(\frac{p_1}{p_2})k+\frac{1}{T_1})^{-1}
$$
Here $k$ is some constant depending on the specific solvent and $T_1$ and $p_1$ is a corresponding pair of known boiling temperature and pressure. For example $100^\circ$C and 1bar for water.
In general the boiling temperature goes down when pressure goes down. (Water boils faster in the Himalaya)

So if you have a typical organic solvent like aceton with a boiling temperature of $56^\circ$C under 1bar it evaporates already at $20^\circ$C when applying a pressure of e.g. 200mbar. (it is not the exact number)
But it also condenses again when cooling down. Using this you can make it liquid again and regain it. Compare this image of a typical distill.
http://www.ssc.education.ed.ac.uk/BSL/pictures/distill.jpg

Summarizing: you use a pump to get into your desired range of boiling temperature and cool the gas in the condenser again to a temperature below the boiling temperature. Depending on your system you have boiling temperatures below $0^\circ$C and cool with liquid nitrogen in the condenser or normal temperature range and cool with water or...