When I sewed with tweed, in clothing, I always had to "underline" each piece, for stability because it was a loose weave. I would use a tightly woven, light weight, muslin fabric of suitable coloring like gray, brown, or whatever enhanced the tweed especially if the weave is so loose the underlining might peak through. I stitched 1/2" from the edge of each piece, having cut the piece expecting to use a 5/8" seam. It was there, to the underline fabric of each piece, that I addressed the fray-ability of the tweed by zigzaging the edges to the underlining. At that point I treated it as any other fabric. I think your quilt could be very beautiful. Three concerns I have are:
1) it will be very heavy, 2) It will be too warm to ever sleep under, hence the picnic idea is good and 3) the cost of dry cleaning and presence of chemicals (and smells) trapped in the batting.
Personally, i would consider making a none-utilitarian, artsy, wall hanging quilt out of the samples that was decorative and a great reminder of your mother and her business.
Whatever direction you go, please share your progress with us, we will cheer you on every step of the way. I know how fabric "calls your name" to use it and every quilt is a challenge and an opportunity to learn, for all of us. Jane