I would keep the quilting that was already done, as it has sentimental value. :)

You can add borders to it, but will have to add batting to the existing and whip stitch the batting to the existing batting, the same with backing, you will have to sew a section of backing to each side.

I have done this on a quilt that I was asked to increase in size for someone when they got a bigger bed. I had no problem with adding batting to it and I did the same with the backing, then quilted the addition to the quilt. :)

Thanks, everyone! What a busy board this is! I didn't expect to get so many responses. I think I will just bind it, as this will be the first major project I've tackled. Also, it seems clear that that's what she intended to do herself, and I agree with the others that it would be nice to keep the quilt close to what she'd planned. I'm definitely leaving her stitches -- I don't think I could bring myself to tear hers out. What do y'all think of adding the X's and O's (or hearts, perhaps?) to the squares, though? Like I said, I'm not sure how the X's will look on the solid-colored backing, since the squares don't match up exactly.

What a wonderful gift your Father gave you :) I too would leave everything just the way Grandma left it and finish it with a solid blue binding. If you wanted to add 'hearts' in a few scattered squares, it would remind you of her and the love she put into the quilt top.