I'm sorry you lost your dad. I would recommend not mixing the fabric types since they will handle differently in the wash, and they have a different weight. One of the patterns that comes to mind is the log cabin.
Do you have any pictures of your dad/family? You could have them printed on fabric (muslin) and surround them with strips of the shirts.
I'm sure it will be a wonderful memory that will last a long time.
You mentioned a few times that you use a "regular sewing machine" - that is what most of us use as well. So don't worry about the sewing part. If you can sew a straight stitch, you can make a quilt top.