[EXTERNAL] Error message: recipe for target 'cygwin' failed

I've successfully built LAMMPS under cygwin many, many times, including with the current version. Here are some things you might try:
1) make sure you've got the latest version of LAMMPS with all bug fixes
2) make sure your source code is clean --- no modifications
3) do a "make clean-all"
4) try starting with "make no-all" --- build with no packages for starters
5) start with the Makefile.cygwin that comes with LAMMPS, then modify as needed
6) re-read the instructions in the manual on building LAMMPS

This should work. If all else fails, find a local expert who can help you with the build process.

I'll also note that we distribute LAMMPS Windows executables, which might be adequate for users who are running on a Windows desktop, don't need to change the source code, don't need every package, and don't want to mess with the build process.