Been there and done that, although I did not have a family at the time. Things on my list:
1. Job hunting and professional development. My employer was famously unstable so I spent some time looking around at other options, networking in on-line forums, etc. I did in fact end up getting a new job.
2. Side hustle--nothing fancy here, I just did online consumer surveys and Amazon Turk. It's not a lot per hour, but since I was going to be home anyway, why not? It accumulated to some real money. Being home during the day, depending where you live, means you might also be able to do focus-groups in person. Those pay much better.
3. Freeing up weekends by getting things done on during the weekday--not just cooking and cleaning, but errands like haircuts, car maintenance, taking checks to the bank, etc. I would think with a family there would be a lot of stuff like that to do.
4. Doing more time-consuming frugal and DIY projects--for instance I liked going to the odd lots grocery store that had super prices but was quite inconvenient (and crowded on weekends). Even if you're not a DIY guru, perhaps there are projects like taking old stuff to the Goodwill or cleaning up the yard that need to be done? I bet if you sat down with your wife and brainstormed a list together you would come up with a number of things to be done. I've got a list of about twelve and I live in a tiny condo!
5. Fun: I actually ended up doing a lot less of this than I anticipated because 1-4 kept me so busy. Some of the things on my list--free lunchtime concerts put on at various venues for the office workers downtown, free library programs, matinees of movies that nobody else wanted to see with me, etc. etc. If you start perusing the calendars of libraries, museums, etc you will see that there are quite a few cheap/free programs during the day because they are aimed at retirees who are presumed to have limited budgets. I did already have a volunteer gig and they were delighted to have someone available during the weekday (it was mostly professional people who volunteered on weekends). Also, protip--if you are the kind of person who enjoys museums, 2-5 PM on weekdays is the best time to go to avoid crowds. The school groups have gone home, the tourists are tired out, and working people are still at their day jobs. If your wife gets an actual lunch break at work, maybe you could take a nice lunch to her and have some couple time.
