Cuidándote is the gerund form of the verb cuidar (to take care of) combined with the pronoun te (you). It means "taking care of you" or "watching over you."
It's an interesting word because it encapsulates a deep sense of devotion and protection, which is a central theme of the song. The singer promises to be there, saying, "No me iré, mi lugar, es aquí cuidándote" (I won't leave, my place is here taking care of you), offering comfort and security to the person they love.
The song is a list of what someone else might need, and the singer keeps himself out of it for as long as he can. "Tal vez alguien con quien caminar" (maybe someone to walk with), "Que le guste tanto el mar" (who likes the sea as much as she does), "Tal vez un amor sin dolor" (maybe a love without pain), "Que no haga daño" (that doesn't cause harm). Every item is framed as hers to want, never as his to offer. Then the verse turns on him: "Una ilusión que tal vez" (a hope that maybe) — "Tal vez pueda ser yo" (maybe it could be me). The hedge is carrying the whole song. He describes the place beside her in detail and then will not quite claim it.
The closing section drops the conditional. "Siente cómo te amo" (feel how I love you), "Toma mi mano" (take my hand), and then the two lines the song has been circling: "Ahora duérmete, aquí estoy" (now go to sleep, I'm here) and "No me iré, mi lugar, es aquí cuidándote" (I won't leave, my place is here taking care of you). A song built on tal vez ends without one.