The confusion arises because we never hear the end of the word, as it overlaps with the next line, that starts with:
So we hear something like:
suaveci-- Y yo
In many Spanish speaking regions including Mexico, Y yo sounds exactly as -illo. That's why one could hear this as suavecillo. See this question and this Wikipedia article
In the release version (2:06) of the song, Anita Tijoux sings the El viento... line, and Julieta Venegas the following one. But when singing live you can hear Venegas recreating this overlapping effect by herself. Like here (2:20), here (2:20), and here (2:33).
If she could finish pronouncing the word she would certainly say suavecito, as she does in her song with that exact title (0:41) :)
As the other answers explained, suavecito is the standard diminutive for suave (soft). It can be understood as even softer than just soft.
As Hithere's answer said, the diminutive can also be form with -illo in some regions, which together with the overlapping effect may have led some people to think suavecillo was what's said in the song.