Apart from cooking, homework assistance, cleaning, or driving (which I assume will be the top things mentioned), here are a couple of ideas that would probably be very successful: car drop off and pick up. Let me explain: say I need maintenance on my car, but tomorrow I am slated to go out of town early in the morning on a road trip (or pretend my tire may have a nail, or I need new tires, so the road trip would be a no go). I work full time, so getting my car into the mechanic is impossible, because he happens to work the same hours I do and I cannot take time off from work. He is not open early enough to go before my road trip. You would come to my home before I go to work, sit with me in the car, drop me off at work, and take my car to the mechanic to get the work done. When it is ready, you would either drive it back and leave it in its parking spot so my car is ready to be driven home, or you'd come pick me up in the evening when I am ready to go home for the day, and then I would drop you off at your home, bus stop, wherever. Same would apply for things like car washes (I don't have time for those as I cannot put my car through a machine and the hand washing places tend to close early), picking up groceries, taking packages to the post office or other little emergencies that may come up, like having to pick up kids from sports during working hours.
Another idea: house sitting...sort of. I live in a condo. Condos have very strict rules that entail moving can only take place Mon-Fri, 9-5, and same goes for any type of labor/repairs/furniture delivery. Someone must be present to open the door and supervise, which is an issue for us working moms. You would stay at home and open the door/supervise furniture deliveries, home repairs or remodeling, installation of appliances, etc., so that we would not have to get out of work or lose a day of work in addressing this. I would love to find someone I could pay to do this, but all my friends and family work, and my mom lives 30 miles away and cannot drive. Having someone who I can trust to not steal and will report to me if the workers damaged my walls, for example, would be great. This would not be exclusive to people living in condos -- people living in homes may only find employees that can work during the week while they are at work, or when they are out of town, so you would take care of that aspect for them.
Good luck with the errand service, whatever it is you decide to do!