I would have friends sign blocks before the top is completed. Have extra blocks on hand. This way, if a mistake is made, it won't have to be in the quilt. Also, if any of the extra blocks are not needed for mistakes, you can use them for a pillow, bag, or other accessory item for your niece.
Make sure to use freezer paper on the BACK of the blocks. This gives a stable surface for people to write. Also, make sure to clearly outline the seam allowances (you could baste 1/4" away from the edge) so that people don't get into them.
If you finish the quilt before getting the signatures, it's okay to iron. However, you take the chance on people messing up blocks and lose out on being able to have the freezer paper stability for writing on the blocks.
BTW, my current avatar is a signature quilt I made for my boss who retired. You can see it better here:
http://www.quiltingboard.com/t-131269-1.htm