How about the Janome Sew Mini? It's a great little machine. It's usually about $49 at Hancock Fabrics. It's small and people have purchased it for traveling or to keep in their motorhome.
One other patch sewing tip (have four girls who were/are Girl Scouts) - sew them on with clear thread in the top of your machine and then use thread to match the shirt on the bottom. It works great and the stitching doesn't show on the top so when you are sewing on round patches, just sew in a square around. I have done this for years and they look great.
Meg