I've always assumed that Alcott used the name
Amy because it's an anagram of the name of the real-life sister upon whom the character was based,
May.
Beth was based upon her sister
Lizzie, and there both the character and the real-life sister had the full name
Elizabeth. But I don't know why
Anna and
Louisa are not at all similar to
Margaret nn
Meg and
Josephine nn
Jo.