A Winchester action would cost you about the same but the gunsmithing cost will change unless your uncle is the one doing the gunsmithing on it. Custom guns aren't hard to figure out, like Scotty said, the weights for most components are out there. To keep it light you'd want a #3 or #4 contour barrel. 26" long would be my recommendation for a hotrod like the 257 Weatherby. After that it's picking the stock and action to use to build it. Most of the components can be found in stock now days.
I just built a 300wsm on a Remington stainless long action with a #5 Brux barrel 26" long plus a muzzle brake in a McMillan Game Scout stock that weighs 10lbs 5 oz with a 6.5x20x50 Leupold in Talley light weight mounts. The bare rifle weighed about 8lbs 6oz. The same gun in a 257 Weatherby with a #3 would weigh about 14oz less or around 7 1/2lbs. The #3 barrel is rated for a pound less but the bore is smaller so it will weigh a few ounces more then the rated difference. A McMillan Hunter stock or one with Edge fill will save some weight but I wouldn't use smaller than a #3 barrel or a #4 fluted. I would also pick a cut rifled barrel if it was me. From my experience as the contours get smaller the cut rifled barrels seem to be more accurate and shift less with heat.
If you decide to go the custom route don't be afraid to ask questions. I can help you or your uncle find parts or a gunsmith. I'm sure others would be more than willing to help as well.
I just built a 300wsm on a Remington stainless long action with a #5 Brux barrel 26" long plus a muzzle brake in a McMillan Game Scout stock that weighs 10lbs 5 oz with a 6.5x20x50 Leupold in Talley light weight mounts. The bare rifle weighed about 8lbs 6oz. The same gun in a 257 Weatherby with a #3 would weigh about 14oz less or around 7 1/2lbs. The #3 barrel is rated for a pound less but the bore is smaller so it will weigh a few ounces more then the rated difference. A McMillan Hunter stock or one with Edge fill will save some weight but I wouldn't use smaller than a #3 barrel or a #4 fluted. I would also pick a cut rifled barrel if it was me. From my experience as the contours get smaller the cut rifled barrels seem to be more accurate and shift less with heat.
If you decide to go the custom route don't be afraid to ask questions. I can help you or your uncle find parts or a gunsmith. I'm sure others would be more than willing to help as well.