I'm not sure about extra income... but thinking outside the box, to my understanding one can save a ton of money by coupon clipping and searching for better deals. It takes time, but could be a good investment. I know there are a ton of websites and online communities dedicated to it. Upon a very cursory internet search:
I started doing one of those online survey things where they pay you per survey you fill out. The pay is proportional to the expected length of the survey, but I found that hourly payment-wise, you'd be better off flipping burgers at McDonald's.
I have no idea if the time you have available (10-15 hours/week?) would be better spent focusing on savings or extra income, but you might try putting it towards researching more cost-effective shopping and such and see if it pays off, i.e. how much money you saved vs how many hours you invested into it.
For part-time jobs though, have you considered food service or something like that? I know you've mentioned wanting to work on your weight too. Waiting on tables will keep you on your feet, your blood flowing, heart-rate higher than just sitting, muscle movement, etc. Typically you can have pretty flexible hours and maybe just work a couple evenings a week. I worked in food service a long time ago and really enjoyed it a lot. I think it's very engaging, fun, challenging, and fulfilling work.