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I think the best approach may depend on how many different pieces and
movement types you use.  One idea may be to use adjectives that are 
mnemonic.  For instance, I am playtesting an unpublished variant of Falcon
Chess with Peter Aronson in which the Falcon move is combined with other
powers.  When an otherwise normal piece also has the Falcon move, the
adjective 'winged' is added to its name.  If there were no more than
four things combined together, a 'Winged Roving Leaping Whatever' might
be easier to remember.
Ralph Betza has taken the approach, in some variants, of trying to
arbitrarily arrange his funny notation to be pronouncable, with mixed
results.  Similarly, you could assign an open syllable to each combining
part, and form nonsense words that would at least be precise and
pronouncable.  So a 'Winged Roving Leaping Whatever' might be a
'WheeRoLee Whatever' or a 'WheeRoLeeWha'.