Hand strength is kind of a big deal, and most people just take it for granted, but when your hands are too weak to do simple everyday tasks, then you definitely want to do some strengthening. There are many tools that are made specifically for hand strengthening, such as therapy putty, gel balls, digit exercisers, etc, but you can do some good strengthening using objects that you have laying around the house.
When looking at what to use for strengthening, you first have to look at the movements that you want to strengthen. You have whole hand squeezing, hand opening, finger pinching, finger spreading, and finger squeezing. With those hand movements in mind, lets think about how to get those movements and tools to use.
Play dough is something that many people have around the house for kids to play with. It is also very easy to make your own out of kitchen ingredients. You can squeeze it and pinch it and get some great strengthening with it.
Sponge / Washcloth
A sponge in a sink or bucket of water is great for whole hand strengthening. Squeezing and wringing the water out of a sponge can be a challenge. Here is a game of squeezing water on chalk.
A bulb turkey baster is great for squeezing and transferring water from one container to another.
The eye dropper has the same concept as the turkey baster, but you are squeezing with your fingers rather than your whole hand. Check out the post about salt painting with an eye dropper.
Clothespins / Tweezers / Tongs / Chopsticks
These tools all work on a similar concept of pinching items to pick the up and transfer them to another container. there is a whole category about clothespins and tongs activities, and another about chopsticks.
Rubber bands are stretchy, so then give a great resistance to the fingers to strengthen them. You can put them over your fingers and spread the band to put it over a container. You can pinch them from each side to place them over a container. This picture is of a stretchy finger exerciser that I made.