What does minage a moi mean?

"minage a moi" doesn't make much sense (literally, it's something like "my mining"). My guess is you're referring to the classic Édith Piaf song, "Mon manège à moi", which means "my personal merry-go-round" or "my own merry-go-round" ("mon" and "à moi" both mean the same thing--"my/mine"). The song is addressing a lover, who metaphorically spins her world around, making her happy, etc.