"eya" has no meaning. I love Nana de la mora in which you can find "ea" instead. The lyrics are by José Ángel Valente (1929-1980) and the music by Paco Ibañez (1934- ) so it's under copyright and will be during our life time. It's on YouTube too. I'm adding this link in case someone is interested in listening other songs by Paco Ibañez.
"eya/ella/ea" is a sort of humming "la la la", yes.

"A la nanita nana" has been translated as "Rock-a-bye Baby" on Hilary Field & Patrice O'Neill's cd Siente. Let's say that "a" being a proposition and this "a" linking "a la nanita nana" to "mi Jesús tiene sueño" with the meaning that Jesus is sleepy because of the lullaby are two different things. Look a the ones on page 85 (= page 86 printed on the document). You also find "a la nana" and this "a" isn't used to link "la nana" to anything.