I have a stash with a bunch of files in it.
But I can't apply my stash because of a conflicting file. I've identified the problematic file in my stash and I want to remove it.
How can I remove a single file from a stash without destroying the entire thing?
A stash is a commit (or really, two or even sometimes three commits) and you cannot change a commit. The literal answer to your question, then, is "you can't". Fortunately, you don't need to.
You say you can't apply your stash because of a conflicting file. But you can apply it, you just get a merge conflict. All you need to do is resolve the merge conflict.
Let's say the conflict is in file
README.txt, just so there's something to write about here.
If you want to resolve it by keeping the on-branch version, apply the stash, then check out the on-branch version to resolve the conflict:
git stash apply git checkout --ours -- README.txt # or git checkout HEAD -- README.txt
If you want to keep the in-stash version, extract that one:
git checkout --theirs -- README.txt # or git checkout stash -- README.txt
Or, use any old merge resolution tool (I just use a text editor), and then "git add" the result.
Once you are all done with the stash,
git stash drop will "forget" the commits that make up the stash. (Don't do this until you are sure you are done with it; it's very hard to get it back afterward.)