Mother, mother, her mother, Enna

Here's something I've been going round and round with.

Hadde is the protagonist of the novel. The entire novel is told from her (3rd person) point of view. Hadde's mother's name is Enna. However, Hadde thinks of Enna as "mother".

One issue... how do I introduce Hadde's mother's name if Hadde never uses it or thinks of her that way? I could just write it, but I feel like that would be an intrusion of the the omniscient author.

And how should I write the following?

"What are you doing, Mother?" asked Hadde.
"What are you doing, Enna?" asked Hadde.
"What are you doing, mother?" asked Hadde.

"I'm saving your life," her mother said.
"I'm saving your life," Mother said.
"I'm saving your life," Enna said.

First draft, it was lots of "Mother" and "her mother said."
Second draft, it was "Mother" and "Mother said."

This draft:
"What are you doing, mother?" asked Hadde.
"I'm saving your life," Enna said.

Do other writers go through this?