I worked this out thanks to a search. The solution is to never create a .resx file with the name "Resources.resx" because it conflicts with the "Resources.resx" files that can be automatically created in the project's properties dialog.
Once I renamed Resources.resx to Icons.resx, I was able to change the image property for anything in the solution.
与恶龙缠斗过久,自身亦成为恶龙;凝视深渊过久,深渊将回以凝视…