Hey Hills in Himachal are safe during rains, you just need to avoid heavy rains.
These areas are fine to visit Jibhi, Manali, Kasol, Dalhousie, Mcleodganj and Khajjiat.
Other than that you can go for a day trek at Triund, Kheerganag, Brighu lake and Chandrataal lake.
Massourie is best place because it is near by Delhi. this the rainy season so everywhere you find rain. But this time is very less crowded in hillstations and you can find peace and enjoy lots, So don't worry and go there where you want to go.
I personally won't suggest you to travel in mountains during rainy seasons. Though Himachal mountains are quite rocky and there are less chances of landslide but still if it rains which is kind of very unpredictable, it may become difficult to explore where you go...
We visited the mountains of Leh Ladakh in the first week of July .And it is said to be the best time for the trip cause the weather is bearable during this period . And what more than the mountains of Leh !!!