Hi Christopher! I think I can help you with recommendations of what to do in Tokyo on a rainy day.
Tokyo has lots of cool indoor activities to enjoy. Last year I had a blast on a rainy weekend with my wife. here's what I'd recommend checking out:
-Shinagawa Aqua park is a lot of fun. Its an aquarium mixed with an amusement park and has animal shows, including an amazing dolphin one. You can spend around 2 or 3 hours there.
-TeNQ space museum is a fun museum about space with interactive panels, games, lots of information, a giant screen on the floor that simulates the world. It's a worthwhile visit. Also its besides Tokyo Dome.
-Go have some fun in Tokyo Dome City. Its a cool amusement park where you can either buy a day ticket or purchase each attraction separately. its fun and if you are already on the rain, you can jump in the water rides too. But if you don't want to get wet, bring your umbrella and enjoy the indoor rides. https://planetyze.com/en/japan/tokyo/tokyo-dome-city
-Go to the Shibuya VR Park. Its a very cool park in the heart of Shibuya where you can enjoy various VR games. its really cool. I jsut don't recommend the scary game, don't waste time on that one, its not scary at all and they use old VR tecnology. But all of the others have a scenery, uses the best VR equipment available and its really enjoyable. http://www.adores.jp/vrpark/
-Go to the Tokyo Art Aquarium. Just watch this video and you'll be interested: https://planetyze.com/en/japan/tokyo/tokyo-art-aquarium
-Go to joypolis in Odaiba. its an inside arcade-theme park where you can spend many hours or just a few enjoying crazy indoor rides and even a rollercoaster. look for it, we've been there and it was awesome.
-Go to the onsen theme park Tokyo Odaiba Oedo Onsen Monogatari. An onsen theme park simulating the Oedo era, where you can thoroughly enjoy natural hot springs in Odaiba. https://planetyze.com/en/japan/tokyo/tokyo-odaiba-oedo-onsen-monogatari
And much more, please ask here if you need more recommendations!