There’s really no good way.
Myself, I change a column name at a time and correct all the errors the change causes. It’s ugly and time-consuming. Note that while the app has errors, users won’t be able to use the app.
Before starting, take note of the latest working version of your app. If you totally break things, you can return to the working version and start over. You can also View that working version in a separate tab or window as a reference, which is especially useful when AppSheet loses an expression because of a column name change (see below).
If a column’s Initial value expression uses a column name that has been changed, the app editor will blank that Initial value expression! Having a reference to the working expressions in a previous app version (as above) is super handy to recover from this.
You could also consider using the App Upgrade feature, which I myself have only recently become aware and haven’t yet tried. It appears you could make a copy of your app, modify the copy, get the copy working the way you want, then replace the live version with the updated copy. This completely prevents user problems while you do your work.
And, of course, you have this option:
You’ll probably find this link useful as you try to track down occurrences of each name:
Also note that some of the name changes you make will be propagated throughout your app by the app editor, but some won’t. So if you change a name and notice it’s been updated elsewhere in your app, don’t be surprised. But also don’t be surprised if the change doesn’t get made everywhere and you have to do it yourself. It’s very inconsistent.