In my own personal fanon, based on my first R/Q fic (in which Rachel and Quinn discussed baby names and Rachel was dead-set on Liza) I've decided that they settled on Eliza as a compromise. Because it still has Liza in it and they both love 'My Fair Lady'. It was only much later that Rachel discovered a copy of 'Bring it on' hidden in Quinn's closet and realised who the baby was really named after ;)

I can imagine Finn walking in holding the containers, but not Quinn. She is clever, remember?