I knew a single mom (she wasn't homeschooling but she had a child who was under school age thus with her all day) who made a living cooking out of her home and selling single serving size pies at a farmers market. She worked hard, was very professional, and managed to eke out a living without having to put her child in daycare to do it. There is also the "personal chef" job if one was in an area that could support it - I can see this as very doable with kids.
The friend I know who does medical transcription does not homeschool - she is pretty consistantly working during "work" hours - I don't know if this would lend itself to stop/start work to help a homeschooler with their lessons etc. It does seem to pay well, and maybe could be done part time (work days, school days). She had to invest time and money into training to get her position, and she is not single, so her husband provided the sole means of support while she pursued the training.
The etsy route is good if combined with other outlets - farmers markets, local stores, craft shows. All of these options are hard work.
I am not a single mom, but I do a csa type produce delivery service that could be done by a single mom - my kids just rode along with me when I did the deliveries. This is the equivalent of a part time job for me. If I had to expand this to full time income I could but it would be alot more work than I want to do. A diviersified income from farmers market, cooking and craft sales could be a wholesome way for a single mom's family to earn a living - kids can help out, not be placed in the position of hindering their parent from earning an income.
I too am concerned about my kids finding a job someday - I am going to start a new thread asking the question "do any of your children earn an income/have their own business?"