In an interview with Vulture, White confirmed the demand - but said she wanted to use half to rebrand herself, and the other half would be donated to organisations supporting independent black artists.
She added that, after speaking with Lady A about recording a new song together, the band sent a contract that she found unsatisfactory.
"It said that we would coexist and that they would use their best efforts to assist me on social media platforms, Amazon, iTunes, all that," White said.
"But what does that mean? I had suggested on the Zoom call that they go by the Band Lady A, or Lady A the Band, and I could be Lady A the Artist, but they didn't want to do that."
White said she had expected the band to take legal action because they seemed "unwilling to compromise" from the outset.
"I think they always knew what they were gonna do," she told Vulture.
According to papers filed in Tennessee, the country band are not demanding money from White and insist she will be allowed to continue performing under the name. The purpose of the case is mainly to assert their rights to the Lady A trademark, which they say they have held since 2011.
Responding to the case, White said: "I was quiet for two weeks because I was trying to believe that it was going to be OK and that they would realise that it would be easier to just change their name, or pay me for my name.
"Five million dollars is nothing, and I'm actually worth more than that, regardless of what they think. But here we go again with another white person trying to take something from a black person, even though they say they're trying to help.
"If you want to be an advocate or an ally, you help those who you're oppressing. And that might require you to give up something, because I am not going to be erased."
The band, who have won five Grammys and had three US number one albums, have not responded to her latest comments.