Two questions about custom naming of glyphs (symbols in this case):
is it possible to customise the naming of glyphs that have an assigned unicode so that the new name appears when you hover over it in the glyphs window in apps like InDesign?
if not, is it possible to customise the naming of PUA glyphs to do the same thing?
The production name is what is used in the exported font file. This is what the glyph palette in Adobe apps shows when the mouse pointer hovers over a glyph.
Actually, are you sure about that? I just checked with some existing fonts that have custom production names for some glyphs, and while those custom names do show up in FontGoggles, the InDesign tooltips consistently do not show them, but the standard Unicode names. Glyphs in the PUA all show <private use area-UXXX>, regardless of their production names.
I don’t have InDesign handy to test, so I might be misremembering this. That would be unfortunate. If you have a client for whom the symbols and symbol names are important, it might make sense to make a document where they can search by name and copy and paste the characters. Possibly a simple HTML file would suffice.