How to make a symbolic link with Cygwin in Windows 7

In short, define the following environment variable:

CYGWIN=winsymlinks:nativestrict

According to Cygwin documentation:

If set to winsymlinks:native or winsymlinks:nativestrict, Cygwin creates symlinks as native Windows symlinks on filesystems and OS versions supporting them.

The difference between winsymlinks:native and winsymlinks:nativestrict is this: If the filesystem supports native symlinks and Cygwin fails to create a native symlink for some reason, it will fall back to creating Cygwin default symlinks with winsymlinks:native, while with winsymlinks:nativestrict the symlink(2) system call will immediately fail.

You should also make sure you run Cygwin with elevated privileges (right-click the shortcut and choose Run as Administrator, or set the mintty shortcut property, Advanced → Run as Administrator).

Some details are provided in the other answer.