Someone make this make sense.
Okay, here it goes. From what I read on twitter, assuming the info people were talking about is accurate:
Lady A has been performing under that name for 20 years. Lady Antebellum has been performing under their name for about 14 years. Perhaps it's because of the four syllable word in their name and the understanding from their legal representatives that an abbreviation might catch on, but Lady Antebllum filed a trademark for "Lady A" about 10 years ago.
After using their existing trademark to change the name of the band, either they then became aware of the solo artist "Lady A," or they've known about her for a number of years. That part isn't clear. Apparently they wanted exclusive rights to use the name, and she requested 10 million dollars. Their counter offer is suing her with the intent that both parties will have legal access to the name, and that nobody will be forced to change again.
So maybe not QUITE as awful as it sounds at first, but still pretty bad. While I'm sure all the promotion for the original Lady A is nice in the short term, I can't imagine a solo artist wants to forever have to explain how she's "not that Lady A." It all makes a bit more sense now, but I still think this is bad move for the band.