What does the idiom 'bought for a song' mean?

Answer 1

It means to have been bought at a significant bargain or with very little effort.

Almost as if you sang to the salesman and charmed him into giving it away (the charming power of song is told in various legends and Fairy Tales).

OR:

I believe that the expression comes from coin collectors or numismatists referring to someone being fooled by a novice who tries to sell him or her for a "song" a copper Chinese coin that was minted about 1,000 years ago during the Song Dynasty. The numismatist knows that literally billions of such coins were produced in the year 1085, for example, in factories across China. They are essentially worthless because of their sheer quantity available - you can buy these 1,000 year old artifacts for less than a dollar today (typically much less).

Hence, "bought it for a song" would refer to the use of real but low value money.