On second look, you are trying to build the cross-compiler, not the OS. My mistake.
Still, the stdio.h in your gcc directory is not the file that the compiler is looking for. The correct file should be in /usr/include. It appears your cygwin installation is missing this necessary file. I'm not very familiar with cygwin, but this SO post
here indicates that you need to install the package cygwin-devel to install the standard c headers.
Wrong info: Presumably you are using a cross compiler to build your OS?
In this case, the compiler will not look for stdio.h in your gcc files, but will expect you to make your own stdio.h and tell the compiler where to find it using a custom include path. This is intentional, as the stdio in gcc will be for your host OS, not for your new OS.