C.,
I have a 25-year-old daughter who was extremely shy, so much so that I asked our family physician for advice when she was about 3. I could not leave her with anyone without her crying the whole time. I put her in a play group with other kids at about 2-1/2, but she still wouldn't really interact with the kids and preferred to play quietly alone. It was hard to get her to speak to anyone. I did all I could to encourage her and, for example, would tell her to go show so-and-so a toy or something and tell them about it. As she got older, things got a little better, but probably not until first grade. All through high school, even while she grew to be outgoing with friends, teachers would comment on how quiet she was. Now, at 25, she sings (even solos), works full time and sells Mary Kay in her spare time, and the joke that a family member once said is, "If she got as much exercise as her mouth, she'd be Super Woman."
Your granddaughter will grow out of it, and the best thing you can do now is to keep giving her opportunities to interact with other kids and adults, without Mom or Dad or brother. As long as a family member is around, she will stick with that person, as my daughter did. See if Mom will put her in a preschool or play group. Have a private meeting with the preschool teacher to develop a game plan that encourages your granddaughter to step out of her comfort zone a bit. It will take time, but she will learn to take baby steps. If possible, have her spend a night or two with you to get used to being away from Mom and Dad. Does Mom have a good and patient friend with a similar-aged child who would be willing to keep your granddaughter overnight?
I too was shy as a kid, but not as much so as my daughter was. While I understood her viewpoint to a degree, it was frustrating because, as a parent, I wanted to see her socialize with others. I think I didn't want her to experience the same feeling I had at times of being an "outsider."
In a nutshell, encourage her and just give her those opportunities to interact with people of all ages. Accept her and love her for her uniqueness, and give her the chance to learn to be independent. It may be a slow process, but be assured that she will be fine. And remember that even as adults, some of us are just more outgoing than others but are still able to be successful in life.
Best wishes.