There are two aspects to the amount of 12V power an inverter will require.
The inverter itself needs some power to run. This is usually only a few watts, so you'll usually be able to turn on the inverter (with no AC-powered devices plugged into it) even if your 12V supply can only provide 1A of current. (A really big inverter may have a larger start-up surge current than that, though.)
Then, on top of that, your AC-powered devices require additional power when plugged into the inverter and turned on.
You may have AC-powered devices that need only a very small amount of power, like an alarm clock or a cellphone charger. These might run fine from a good inverter connected to your 12V, 1A supply. Or you may want to run a microwave oven, which will obviously need a lot more power than your supply can provide.
This is why I suggested using your generator and solar cells to charge a battery, because the battery will be able to provide larger amounts of power for short periods of time. The low-cost 12V, 7AH gel-cell battery I mentioned before can provide roughly 7A or 90W of output power for about half an hour-- plus or minus, depending on condition-- even if you can only charge it at 1A.
Now, 90W isn't enough to run a microwave, but it would be enough to run a small TV or a laptop charger for a while, so you may find that useful. Adding a battery and a battery charger circuit will add cost, size, and weight, but greatly increases the functionality of the solution. So it's your call.
Best,
. png