How do i go about changing my middle name?

i went to my local dmv in N.Y to get my non driver i.d. the lady said she couldn't do anything for me because my social security had breyanna joynae' celestin & my birth certificate has breyanna ioynae jacqueline celestin. so how do i go about fixing this ? i just want "Breyanna Joynae Celestin" . I need to fix my middle name asap so i could have i.d to cash my checks etc ..