Another of those phrases (like "make groceries") that has crept into my vocabulary from grandparents. This one from my maternal grandmother. A dodo chair is basically that soft, comfortable chair that you sink into that seems to have the magical ability to make your eyes heavy within minutes. It refers to sleep, not the bird. One of my kids thought for years that it meant it turned you into a dodo. Not too far off the mark if you have a really good dodo chair. Think Sunday afternoon football games snoozing in the chair.
From Wiki: Fais do-do is a name for a Cajun dance party, originating before World War II. According to Mark Humphrey's notes from the Roots n' Blues CD "Cajun Dance Party - Fais Do-Do", the parties were named for "the gentle command ('go to sleep') young mothers offered bawling infants." He quotes early Cajun musician Edwin Duhon of the Hackberry Ramblers, "She'd go to the cry room, give the baby a nipple and say, 'Fais do-do.' She'd want the baby to go to sleep fast, 'cause she's worried about her husband dancing with somebody else out there."
'Do-do' itself is a shortening of the French verb dormir (to sleep), used primarily in speaking to small children. Comparable to the American English "beddy-bye", it is still commonly used by French-speaking people.