Chuck Hawks' post about the Winchester Model 88
HERE is a good starting point. As the .308 is the most common, value is best determined by it's condition. I've seen pristine .308 versions like mine go for over $1000 without a scope here in Denver.
If your great grandfather bought it around that time, you've got an early version as production started in 1955. Records at the factory indicate the following serial numbers were assigned to guns at the end of the calendar year so you should be able to date yours from the listing below.
MODEL 88
1955 - - - - - 1 TO 18378 . . . . . . 6? - - - - - - - - - 139838 . . . . . . 69 - - - - - - - - H239899
56 - - - - - - - - - - 36756 . . . . . . 63 - - - - - - - - - 148858 . . . . . . 7U - - - - - - - - H258229
57 - - - - - - - - - - 55134 . . . . . . 64 - - - - - - - - - 160307 . . . . . . 71 - - - - - - - - H266784
58 - - - - - - - - - - 73512 . . . . . . 65 - - - - - - - - - 162699 . . . . . . 72 - - - - - - - - H279014
59 - - - - - - - - - - 91890 . . . . . . 66 - - - - - - - - - 192595 . . . . . . 73 - - - - - - - - H283718
60 - - - - - - - - - 110268 . . . . . . 67 - - - - - - - - - 212416
61 - - - - - - - - - 128651 . . . . . . 68 - - - - - - - - - 230199
I've got a 1959 Winchester Model 88 in .308 and with the Winchester Model 1873, I believe that they are arguably the two most beautiful leverguns ever made. Even though the '73 is renowned for it's short, smooth action, my 88's action (unfired when I got it in 2012) is surprisingly smooth even though it has a long throw. It's lines, very reminiscent of my 1951 Winchester Model 70 in 30-06, give it a sleek and timeless look.