If the quilt is old enough for your Great-grandmother to have made it, then almost certainly the ties are silk. Vintage silk has its own issues, but like other fabrics from the same period, silk is suseptible to the chemical dyes.
Those early chemical dyes seemed heaven sent in that they were finally permanent and less suseptible to fading. Unfortunately, some of the metals and elememts they used to get those lovely (especially) purple shades were actually chemically breaking down the fibers. Easy to see why the color purple was traditionally reserved for royalty.
And because silk is normally a relatively soft and drapey fabric, there was also a process used in silk for ladies dresses and crinolines to give it that very sensual and alluring 'rustle'. 'Member the red crinoline that Rhett Bulter gave Mammy? For as repressed as the Victorian era was supposed to be, Madonna and Brittney Spears ain't got nuttin' on our great-grandmothers when it comes to sex appeal! LOL
I suspect that the problem fabrics in your quilt are either purple or were little bits of left over dress silks, or possibly even cigar silks.
Here are your choices, Jenell, as I see them, in no particular order:
1) Replace the damaged fabrics. Since you say the muslin foundation, and the backing are intact, it is possible to carefully remove the damaged sections and, using a tiny whip stitch, to lay a new piece of fabric in and secure in place.
I prefer to use vintage fabric when I do this, but since you aren't even sure when the quilt was made, it will be next to impossible to find silk, much less men's ties from the period. There are lots of reproduction fabrics available in cotton, so new fabric can be substituted if necessary in a cotton quilt.
I am not aware of anyone reproducing vintage silks. Since this quilt is loosely in the crazy quilt genre, how would you feel about using velvet to replace the fabric? Silk and velvet were often used together.
2) Stablize the questionable areas with fine tulle. This is one of the methods used by textile conservators and museums to protect fragile fabrics from further damage.
If you wanted to display the quilt on occassion, you could fold it in such a way that the damaged sections are not visible and drape over the back of the couch. If you want to store the quilt, I can hook you up with some sources for museum quality conservation supplies.
3) If you feel that the damage is too extensive to warrent that much effort, but you don't just want to put it away, you can 'recycle' it. Is there a section large that you could cut it down to a wall hanging for your bedroom or living room or displayed on the back of the couch?
I actually have a couple of vintage quilts where the damage is confined to two sides. My plan is to cut them down and use as wall hangings in the bedroom.
If the undamaged sections are smaller, maybe you could cut smaller sections and have them archivally framed. A series of three or five framed pieces along the back of a couch or going up a stairwell makes for a very interesting conversation piece. If there are several pieces availible, framing them or even making them into to small pillows makes a thoughtful way of sharing grandma's artistry with other family members.
I hope this is helpful. You have quite a treasure and I know you will cherish it, so at the very least, I would suggest you stablize the damaged sections with fine tulle and then clean it.
You will need some nylon window screen, a square however wide the roll is. Bind the edges with a soft (not stiff) durable fabric such as brushed denim, a sportweight gabardine or duck, or even a soft canvas. You will need a large table to work on. Lay the quilt out (take care not to let it dangle over the edge) and place your screen over it. Have another person hold the quilt and screen in place while you carefully vacuum through the screen. Reposition the quilt and screen as needed until the whole surface is completed. You will be amazed how much dust and dirt you can get out this way. If it makes you feel more secure, you can also rubber band some cheesecloth over the end of the nozzle. Just be sure not to reduce the suction too much.
I would also suggest that you document as much as possible about the quilt, including pictures. If you decide to keep the quilt intact, you might also check to see if any of the museums in Oklahoma host a repository for a quilt project. They may be very interested in your documentation.
Good luck, sweetie. Done't hestitate to ask if you have further questions.