A hyper-active dog is going to need a lot more than just a few basic activities to get their energy out. So, a really great option is to just grab one of your dog's favorite toys and start running! They will love chasing you around to try and get their toy back. Another great option is to use a laser pointer to do the work for you. This may not work in a smaller home, but if you live somewhere with a bit of space, chase is the perfect game to get some of those rainy day jitters out.
If your dog knows the 'stay' command, a game of hide and seek is an amazing way to kill the time! Just have them sit in one room and stay while you go find a hiding spot (you can also have a friend or spouse hold the dog). Don't be discouraged if your pet doesn't know the 'stay' command. Staying indoors due to a storm can be a great time to teach your dog new tricks, so they can learn to stay and much more while the rain pours down.
Rainy days are the perfect time to pull out those brushes and get your pet looking fabulous. Obviously, you won't want to take them outside right after, as all of your hard work will be washed down the drain, but during the rain storm there is nothing more relaxing (and bond inducing) then a good brush and pet session.