I remember Linda Lee Danvers very well. Especially when she was elected Governor of Florida in an alternate reality story.
Also, do you think they could ever say they had spoken with their distant cousin, Carol? "wink"

It would seem strange to change her backstory to differentiate her from a much less famous version of the character.
That depends on what you call as 'less' famous version of the characters. The Linda Danvers/Supergirl series by Peter David was and still is the biggest selling and longest running series of Supergirl in history according to DC and if it wasn't for the sexism at Warner Bros in the early 00s that prevented the movie going through 'cause they "didn't want to have an edgier Supergirl than Superman" (Peter David's exact words) I doubt Kara would be Supergirl now. Linda Danvers was far more popular than Kara and the series got more letters from female comic book readers than any other series apart from Wonder Woman. The fact every single Supergirl series with Kara before and after the Linda Danvers hasn't been able to last half as long as that series and the only reason Supergirl is second biggest selling female superhero is 'cause of the sales DC made with the Linda Danvers series speaks volumes about the popularity of Kara.
Gary Frank even said the only reason Linda Danvers wasn't brought back was 'cause Peter David and Geoff Johns had a falling out.

A note to homophobes: Don't hate me because of my sexualityHate me because you don't have the guts to accept yours

This message has been deleted.

Yeah she wasn't kara danvers till much more recently. So they could have changed it

I had to look it up but Kara's human name was Linda Lee until she was adopted by Fred and Edna Danvers and it became Linda Lee Danvers (Action Comics 279, July 1961).

But it wasn't kara danvers it was linda right?

Kara has has Danvers as her human last name since 1961. It would seem strange to change her backstory to differentiate her from a much less famous version of the character.

I know there was a supergirl named linda danvers. Don't care about the specifics as she is often reinvented. I'm saying that since the linda danvers and the kara supergirls are completely different then I wish she had been given q different last name. But because she was given the last name Danvers in this series they have given her the name Danvers in the current Supergirl comic since rebirth (all of 6 issues ago)

No. I think you are getting mixed up with the alias, Linda Lee Danvers. Linda Danvers was the second Supergirl and a completely separate character from Kara.
Linda was a former drug addict and a member of a demonic cult that killed people and sacrificed them to demons and when her ex, Buzz and the rest of the cult turned on her and tried to sacrifice her the Matrix saved her by merging with Linda Danvers and making her the new Supergirl. In the Linda Danvers series Supergirl fights witches, demons, vampires, monsters, aliens, werewolves and even Lucifer to redeem her darkened soul. Her powers while somewhat similar to Superman included the ability to shapeshift, heal people, teleport, shoot psi blasts (telekinetic energy) and see the last few seconds of a persons life by looking into their eyes. She also had Wings of flame when she became the Earth Born Angel of Fire and she was a lot darker, edgier and unpredictable than Kara.
The Linda Danvers/Supergirl series was the biggest selling and longest running series of Supergirl to date and as a female superhero series was second only in sales to Wonder Woman. It also got more letters from female comic book readers than any other female comic book series including Wonder Woman and it is currently being re-released in graphic novel form now.

The catacombs
And revolving doors inside of my brain
Preachers and whores
And the neon pulpits burst into flames
I come from both
Sides of me and either side of their tracks
Humbling my ghosts
Letting life that floods my veins bring me back
The unshackling of the chains on my wrists
And the loyalty to pain that resists
And the greatest books are talking about this
Why must everybody die to exist?