Can you tell a story with just one letter...one word?

You might have better luck with NO notes, as per John Cage's 4'33".

When you say one note, I'm going to assume you mean not just one note played a single time, but one pitch that could be played multiple times. I'm not sure that any one note (C vs E vs G) is better than any other.

As for what to play, I would experiment with playing the same note across different octaves and against different inversions of the C chord. Maybe have the single note of your choice "chase" the chord around. You still have rhythm and volume to use so you could do something with that. I suspect one could do some interesting stuff with poly-rhythms and so on. Maybe play distinct but overlapping rhythms in different octaves that collide and seperate and alternately reinforce or negate each other.

And last, if you wanted to tell a literal story, you could use the note of your choosing to play a series of morse code dots and dashes which spelled out the text of any story you wanted.