i have the same problem in my back yard. maybe you could put in some ferns and hostas? otherwise till it up get rid of the moss and try sod. tell the sod farm its for a shady area. check your ph. that could be the problem. you might have to ammend the soil first.
You might also want to consider cutting down the size of your lawn by adding some beds for plants that like shade or moist soil. Hostas come to mind for shade, and monarda for moist soil if there is some sun.