My first sewing machine was an old Singer Treadle. My sister still has that machine. I was 9 when my dad bought home the machine. He paid $3 for it at an auction. Now mind you I know you are not likely to find this kind of deal today. I have been teaching my now 9 yr old granddaughter with a Featherweight machine. That is a great machine for teaching kids on but not always practical unless you already have that machine.
I would check at a thrift shop for a used machine to start with. I would not invest much only because she may not want to continue sewing and a lot of money can be spent on sewing machines only not to have it used.