I think this was because the two shared the name Laura. I think she used "Beth", which was a nickname Laura was given by Almanzo from her middle name Elizabeth, to distinguish herself from the other Laura. The name 'Beth' started off as 'Bess'.
When she and Almozo first met, she told him Charles called her half pint and she hated it because it made her feel like a kid. Almonzo said he would give her a nickname only he would call her , he came up with Beth.