Goa in Winter is synonymous with a beach holiday, fun and entertainment. Goa in Monsoon is a completely different experience. Beaches are deserted, its green, rains are intermittent heavy to very heavy and countryside is lovely with waterfalls, mist and greenery. Its also the time to try different options.
A Heritage Home - There are number of heritage type Homestays / small resorts in Goa that offer charming stays. Some of these are not on or around a beach and as in Monsoon beach is never a focus area its a good idea to opt for one.
Goa Luxury Resort - Monsoon is lean season period for luxury resorts in Goa and therefore offer good deals. A lot of guests find the best time to take advantage of good offers and enjoy a luxury getaway.
Driving to Goa - Mumbai - Goa - NH17 Coastal highway is legendary in terms of its beauty, greenery and rural feel. It comes alive even more in monsoon. Even the route from Pune via NH-4 is green and beautiful and driving enthusiasts, bikers like it especially in Monsoon.
Experience Extreme Monsoon - For those who haven't had a taste of extreme monsoon, a two night stay in Chorla Ghat or Dandeli would be good. These could turn into 100% humidity zones with nothing drying up. Most often all one can do is to sit around, enjoy some reading / soft music / good company / a drink / complete solitude and admire the nature.
Countryside - Waterfalls and mist is all around the lush green countryside. Dudhsagar falls are at its peak and even Jog falls in Coastal Karnataka is at its peak and something one can access from Goa.
Temples - During Winter Beach and activities take away focus from the fact that there are some significant temples around Goa that are worth a visit. Monsoon is a good time to visit them with lovely drives around the countryside.
Group Fun - During monsoon hotel availability is ample and tariffs are much lower and cost to access Goa (i.e. flight fares) are much lower. So April - Sept is season for groups in Goa. However best time during this period is Monsoon (Mid-June to Mid-Sept). Be it conference, leisure groups, corporate incentives etc. Goa is a popular choice during monsoon.