One of the Beatles first published songs, but an awkward phrasing. Was “Love Me Do” a common English phrase back in the day? The opposite term, “Love Me Don’t” also doesn’t make much sense. Just poetic license?
Pure speculation: there was also, at around that time, a petal plucking refrain (ie, pull a petal from a flower, say a line, pull another petal, say a line) which ran: