You need to use a fusible stabilizer (also called interfacing) like is used for T-shirt quilts. You can get it at JoAnnes or Walmart. Here is one I have used:
http://www.walmart.com/ip/Pellon-Fus...-Bolt/21976551
Iron on wrong side of clothing. do a larger piece than what you want to cut and cut down AFTER fusible has been attached. Then proceed as you would with the rest.
Word of caution, I have made a memory quilt before on commission. It involved mixing fabrics and types. While it is doable, it was most definitely NOT a fun project and I would never do another. You are emotionally vested so you might not find it as frustrating as I did. Mixing various weights of fabric does not always go well with piecing. Be prepared for not all the seams lining up and a fair amount of fussing with it. Also be careful of applying too much heat to certain fabrics. Some knits will melt with too much heat from the iron, depending on what they are made of.