Here's a cheery quilt, made in the late 1930s or the 1940s. The pattern name is Friendship Dahlia.
There's an overall quilt pattern called Dahlia, very complex and not a beginner's quilt by any means. Maybe this block got its name because it's much, much easier and friendlier to make!
Anyhow, the buttonhole stitching holding the petals in place was wearing out. The fabric is nearly all still intact, thin but present, so the restoration was just a matter of lots and lots of buttonhole stitches.
Here's the block with the flapping petals pinned into place.
Here's the buttonhole stitch how-to. You can find full directions in any embroidery stitch book. It's a very basic and very common stitch. I used 2 strands of floss.
And here's the finished block. Using black thread on a pastel quilt was very much the style in the 1920s-40s. If you are doing restoration on a piece from this era, I think it's good to stick with that color scheme and stay true to the sensibility of those times. Deciding to replace the stitching with threads to match the colors of the quilt would add a late 20th - 21st century aesthetic.
FYI: People always want to know how much time things take. On this block, I replaced just the stitching that was missing, along probably about 3/4 of the edges. It took about 2 1/2 hours. Over the entire quilt, with 4 blocks this bad, 4 with a petal or two that needed help, and many blocks needing an inch here and there, I used just over 2 skeins of floss. Total time on the job: 15 hours.