I stopped by Stonewall Jackson Shrine (aka Chandler's Office Building) today and the park ranger there was awesome with sharing many stories with my wife and me. I asked about the reason behind the name of Stonewall Jackson's Shrine since I remembered this question from here on CivilWarTalk.
The ranger said that was a great question and explained in details about it...
There was a girl named Lucy Chandler (12 years of age) who was a daughter of the Chandler family who owned the land. She was watching the situation as it unfolded with Stonewall Jackson in his makeshift hospital room at Chandler's Office Building.
When Jackson passed away, Lucy was distraught and walked to her mother to tell her that she wants to trade places with Jackson because only her parents would be upset if she died, but the whole South would be upset if Jackson died. (I was blown away by this. Amazing humility and intuition for a child.)
When she became older (18 years or so), she kept the clock, blankets, and some relics and kept a diagram of what it used to look like when Stonewall passed away. The doctor took the bed and stored it in his basement then donated it to the museum in Richmond.
In late 1920s, Lucy made a deal to return the relics if the museum will return the bed that Stonewall died in to the Chandler's Office Building. This is when Lucy set things in place according to her diagram (and from her memory) then added the flags of the Confederacy on all four bedposts and on the wall to mark it as the shrine to the Confederacy and General Jackson, hence Stonewall Jackson Shrine. "Shrine" was defined as "a site hallowed by association with a revered person or object or with an important event", rather than the modern definition relating with religion.
When NPS received the land from the railroad company who used to own the land, they decided to remove the Confederate flags from the bedposts and the wall from the historic "shrine" to make it less offensive, due to various perspectives of its symbolism. However, the name "Stonewall Jackson Shrine" stuck ever since since that's how Lucy named it in the first place.
I don't have any sources to back up what I just typed, but hopefully, that is close to how I recalled the ranger's answer to this question.