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Pi filters are often used to block high frequencies between a device and the power line it draws power from.  In such applications, the considerations specifying the inductor include:<ol>

<li>How much power current needs to pass, preferably unimpeded by the filter.

<li>What frequencies need to be attenuated.

<li>How much the stop band frequencies need to be attenuated.

</ol>

If the filter needs to pass 50 or 60 Hz power substantially unimpeded, two things must be true:<ol>

<li>The inductor does not saturate a the peak of the current waveform.  This peak is sqrt(2) higher than the RMS current.

<li>The series resistance must be tolerable.  Not only must the voltage drop across the inductor be "small" compared to the line voltage, the heat power from the resistive loss must be managed.

</ol>

The inductor core material choice is a tradeoff, like it is with all inductors.  Since the current is usually substantial, you need something that can store a lot of magnetic energy in a small volume.  Unfortunately, the best materials for that, like iron, are also conductive.  Ferrites are not conductive, but can't store as much magnetic energy per volume before saturation.  Air never saturates, but the energy density is also much lower.

Then you also have to consider what happens at the frequencies you are trying to attenuate.  Inductive materials aren't perfect, and can stop being inductive above some frequency.  There are lots of different ferrite formulations alone, and then there are laminated iron, and more.

It's OK to have loss at high frequencies, even desirable for a power line filter, but you want to avoid loss at the power frequency.

There are many many tradeoffs.  Too many to give all the details here.  Look at some inductor data sheets to see the various parameters that are considered in their design.