Alwar. Go and experience Dessert Safari in Sariska National Park in the evening!
And you can opt to stay at Siliserh Lake Palace and can visit that place for boating and sightseeing.
Best suited for this weather are nearest hill stations like Corbett, Lansdowne, Almora, Ranikhet, Kasauli, Macleodganj are good and budgeted options which can be covered during weekend.
Monsoon Destinations Around Delhi – Within 200 to 500 Kms
These are the places to visit near Delhi within 200 kms, during monsoon:
Agra – 210 KM
Corbett National Park – 230 KM
Lansdowne – 254 KM
Morni Hills – 272 KM
Parwanoo – 281 KM
Mussoorie – 286 KM
Kasauli – 298 KM
Mandawa – 233 KM
Kangojodi – 276 KM
Landour – 302 KM
Pragpur – 412 KM
Shoghi – 331 KM
Almora – 363 KM
Bundi – 485 KM
Orchha – 498 KM
Dear fellow traveler, for you Lansdowne is perfect place to visit. Lansdowne is just 255 kms from New Delhi and it takes almost 6 hours to reach. Nestled in the lap of the Shivalik Ranges of lower Himalayas, Lansdowne is known for it's scenic beauty, weather and peaceful surroundings. The best time to visit Lansdowne is during Monsoon, during the rains the town became more beautiful and scenic. Here you can also do activities like trekking and camping.
For complete details about Lansdowne check the link below.
https://www.tripoto.com/trip/lansdowne