As @robertferanec says on top it's difficult to write "one answer" because this argument is very large a also very dependant on the application.
The power, topology of the converter and enviroments affect a lot the choice for an inductor.
A really good starting point is to see the documentation about the converter and evaluation board (if avaible) that often both report some good explanation about the design process in order to understand how the converter and topology works.
By the way i'll try to condensate some points that i take in mind for an inductor:
1 - I check the maximum RMS current and peak current and i choose, at the beginning of the design, a value at least two times higher. For example if the calculations gives 2 A for peak current i'll start with an inductor that have 4 A.
2 - I estimate by calculation and/or simulation (especially in high power application) the temperature rise for the inductor based on the maximum current and the DC resistance and i'll check that is reasonable value. Very often is usefull the graph from the manufacturer that plots the temperature rise VS current, but also very often do not forget that yout application could be different, or very different..., from the case that manufactor do the tests...
If the value obtained is too high i'll re-iterate the process with another inductor. Tipically with lowering the DC resistance and/or choosing a "bigger" inductor will help for the temperature.
By hand i never take in account the AC resistance for the inductor but in some application is not negligible.. i've designed a power inductor with 6 Watts dissipated for DC resistance and again about 3 W for the AC resistance! It was an inductor wounded on ETD39 with gapped 3C90 core.
3 - I check the Self Resonance Frequency about the inductor that tipically should be at least 2 times higher than the maximum operation frequency for the inductor. Take in mind that in some cases the operation frequency about the converter, i mean the switching frequency of the active devices, could be different from the frequency that the inductor will work... it depends on the topolgy and application for that inductor. Important to notice that the SRF, basically explained, rappresent the frequency that above the inductor.. do not work as an inductor
4 - I always try to find and inductor with shielded construction if is possible, they will work really better for EMC and EMI performance.
Other things to take in mind for and inductor:
For higher frequency the better result should be with smaller inductor. Especially the height value is very important for this.
For very high frequency application please consider to realize the inductor by the combination of PCB and ferrite cores. I mean do the winding with a track on the PCB copper.
Smaller inductor is also suitable for space limited application and in cases when the PCB could take sever mechanical stress and/or vibration, like industrial enviroment, avionic and so on..
If the temperature rise up, the inductance (tipically) goes down togheter with the DC resistance, so this means that the temperature rise up again...
and we know, in some design the inductor is very very hot, hotter than the active components...
Based on the topology of the inductor i'll try to choose the best core material composite. For example in a boost topology is better (tipically) to have and inductor with smoth saturation inductance. The manufactor provides some graph and parameters to help this kind of choice. Some times they provide also a particular documents for this specific.
Think and spent time in component placement, especially for the inductor. Try to place in a PCB area that can receive a sufficient ventilation. Also try to place it as far as possible from other components that suffer the hot temperature like Crystal and/or Oscillator, E caps, tantalum caps, super caps, back-up battery and soo on. This will help to achieve the best performance for the entire converter (and project) for thermal and electrical sides. It dosen't have much sense to choose an expensive inductor if then is placed in bad ways...
Do some market research in order to have an idea about the price, avaiability and also alternative inductor from two or more manufactor. It's still without much sense to choose an inductor that then have 10 weeks for lead time and other manufactor don't do an alternative solution.
Last but not least, do the test! Test the inductor a lot! In all the worst cases for the converter. This is really the key for the inductor and rest of magnetic parts!
If is possible build up the converter, load it and do all the test with different inductor.
I hope this kind can help, regards.