Playing Beatie Bow by Ruth Park
Worldcat.org abstract:
Disturbed that her mother could welcome back her unfaithful father,
Abigail Kirk undergoes a mysterious voyage to nineteenth-century
Australia, where her experiences help her to understand the power of
love and to accept her father.
The character born Lynette changes her name to Abigail:
'Tell me some witches' names, Mum,' she said.
'Well, there's Samantha, and Tabitha,' Kathy began.
'Oh, I don't want soppy TV names,' said her daughter. 'Some real witches' names.
'They'd have to be old ones,' said Kathy thoughtfully, like Hephzibah, or Susannah, or Petronella, or Abigail.
'That's the one!' cried the girl.
'But it's so plain, so knobbly, so... so awful!' wailed Kathy.
Grandmother smiled. Abigail could see quite easily that Grandmother thought she was plain and knobbly and awful, too. So that settled it.