In order to do that I need to over-provision my solar panels. Solar panels remain a lot cheaper than batteries, so for me at least it makes sense to get a lot of solar panels and only a few "hours" worth of batteries.
Let's assume that some sunny days I have an extra KW of power generation for ~8 hours. On overcast days I probably don't have any extra power, unless I'm not at home and can shut everything down.
What can I use the waste electricity for?
I've bought two different solar systems in two different states and the devil is in the details for your hypothetical situation. You need to understand average overcast days, ~90 percentile run of overcast days in your lowest intensity sun month, intensity of sun, angle of the sun, et c. Also, if you optimize for the worst days of the year, you could be reducing total energy output because your panels would be optimized for a lower sun angle.
You can save a surprising amount of money by just running the most energy intensive appliances during peak solar.
Energy storage, whether purchased batteries or built gravity storage is likely most useful.
Having consistent access to power through predictable reserves makes energy way more useful to you. You can do laundry at night, or run a heater, etc.
Storing energy allows you to adapt your power usage to your needs and preferences, instead of adapting your needs and preferences to power availability—which has daily and seasonal limitations.
You can't. That's really the end of it.
Lots have tried and failed. You can pick a fun project that you might get some skills from but it's not really a real answer.
It won't beat batteries which you've already spec'd out.
Best you have is move chores when you have extra electricity. Washing, download torrents, watch TV, run the pool pump, heat water, dry food.
Not sure if the economics would work out in your favor² but I personally consider going off-the-grid worth the cost.
¹https://www.tesla.com/powerwall
²subsidies: https://www.tesla.com/support/energy/powerwall/learn/incenti... (could potentially knock off 50% of the cost if you qualify)
Let water flow out of it and power a turbine during times where electricity is needed.
Make hydrogen (H2)with electrolysis. Need to found a good technology for that, and be careful hydrogen is very explosive.