Oops, one more part of the cygwin patch (SF patch #102409 by jlt63:

gvanrossum · gvanrossum · commit fef124346ecc · 2001-01-10T21:17:27.000Z
Cygwin Python DLL and Shared Extension Patch). Add module.dll as a valid extension. jlt63 writes: Note that his change essentially backs out the fix for bug #115973. Should ".pyd" be retained instead for posterity?
diff --git a/Python/dynload_shlib.c b/Python/dynload_shlib.c
@@ -29,8 +29,8 @@
 
 const struct filedescr _PyImport_DynLoadFiletab[] = {
 #ifdef __CYGWIN__
-	{".pyd", "rb", C_EXTENSION},
 	{".dll", "rb", C_EXTENSION},
+	{"module.dll", "rb", C_EXTENSION},
 #else
 	{".so", "rb", C_EXTENSION},
 	{"module.so", "rb", C_EXTENSION},
