It's not an ayu song actually - I'd request "ayumi (Steps)" by Aqua Timez. The song in general is about encouraging an insecure person with an independent streak to remain independent, and the lyrics remind me of how Ayu seems to feel about herself & her career.
Plus, y'know. The title.
The name "ayumi" means walking one's own path and I think it's a name Ayu has always tried really hard to live up to, so whenever the "ayumi" concept comes up, it tends to remind me of her anyway.