A lot of places have less crowds when it's raining.
When the sun is shining and the weather is perfect, places like Disneyland or DisneySea for instance are basically wall-to-wall people in the summertime. A rainy day tends to reduce some of the crowd numbers - so as long as you dress appropriately for a rainy day, you won't be grouchy and you'll be dealing with significantly less queues. One year in June, my family and I visited Kamakura to see the Big Buddha - and we managed to get family pictures with basically nobody in the background. Everyone has asked us how on earth we managed that - and I tell them that it was the rain keeping everyone else at home or doing indoor activities!
Everything is a beautiful, lush green - and there are some gorgeous flowers in bloom!
June was actually the first month I experienced Japan (I'm almost coming up on four years living here). I remember thinking as soon as we got out of the airport that everything looked so beautifully lush and green. Also, one of my favorite types of flowers (hydrangeas) are in bloom in June, and that's another reason why I welcome the rainy season with open arms!
The temperatures are a lot more manageable than the full heat of summer
Everyone who has lived in Japan for a while tends to mention how hot and humid the summers get. Rainy season is a chance to experience summer in Japan before it starts getting super hot. Since I'm the kind of person that burns very easily (thanks, red hair and fair skin!) I find it a lot easier to be out and about in the overcast rainy weather than I do when it's super hot and sunny.
Something about the overcast skies makes the colors in pictures really pop.
Japan has a lot of tourist sites that have amazing colors - the Golden Pavilion, loads of shrines and temples with red torii gates, flowers of all different hues - and when the skies are grey it seems to make for amazing pictures. On one occasion that I visited Golden Pavilion it was rainy and overcast, and it made the photos I took look like paintings. You'll have great photos to remember your trip by if you're heading to Japan at this time of the year.
It's an excuse to buy a cute umbrella!
This is serious! Japan has some of the best umbrellas ever - think of cute character umbrellas, beautiful traditional parasol type ones, or (my favorite) umbrellas that display a pattern on them once they get wet! If you're in a pinch and need something functional but basic to keep the rain off you, just about every convenience store will keep simple plastic umbrellas in stock - setting you back only a few hundred yen.
Hopefully those are a few reasons for you to embrace and not fear the rainy season - it really does have several reasons that make it a great time of the year to pay Japan a visit!