You might want to rephrase 'once for each differently named Action card you have in play other than copies of itself' to 'other than this'; the effect (that you only play the chosen card once unless you've played an action to playing Unnamed Card) will be the same, and will be less wordy.
As for the name, perhaps Troubadour, since a Troubadour is only interesting if he knows a wide variety of songs?