What are your skills? What is your experience?
My husband has worked at home as a freelance copywriter for decades. I work at home doing some education in a specific field (nutritional epigenetics), some Spanish tutoring (just to keep my skills sharp), and also doing some freelance editing/proofreading. We both get some work through Thumbtack, which has some up-front fees (you buy credits to use to bid on jobs) but they aren't excessive, and you can choose what to bid on. If we get repeat clients, we don't pay for subsequent jobs for them, so it's very reasonable. I can work the schedule around my other activities, BUT I have skills in these areas, references, etc. I don't know what you know how to do or what your background is.
Data entry won't pay you what you think. Things like medical coding have deadlines and educational/training requirements up front, and you have to work uninterrupted.
Phone work (making appointments, etc.) is impossible to do if you have a child at home. I know people who do this, but their kids are in school or in daycare. Period.
No one will pay you to work while you watch your child and certainly not while you supervise his schooling on top of that. Teaching is a full time job! If you are developing the curriculum yourself, you have more hours to put in to make sure he meets state guidelines. You shouldn't even have time to throw in the laundry or put some stuff in the crockpot to get dinner ready! If you can team up with another homeschooling parent, maybe you can do small classes of 2 kids, and divide up the work giving you some free time, but you still need a job you can structure around that - and very few clients want to wait while you do your lessons. I wouldn't think that providing in-home daycare would work, because you'd be too distracted to provide teaching time to your child. If I had a toddler and needed daycare, I'd be very reluctant to put my child in a home where the daycare provider was also trying to teach an older child - I'd be concerned that one or both kids wouldn't have your undivided attention.
Selling things (which you say you don't want to do anyway), or doing something like sewing/crafts still mean that you have to be out of the house meeting customers, doing craft fairs, and so on, and you really won't have time to create enough inventory to make it profitable. Those jobs depend on volume, and you're looking to do something very sporadic and occasional.
Good luck!