bodyboarding requires a ridiculous amount of flexibility to be in the proper bodyboarding position (riding high on the board, and arching your back up)... mostly in your lower back, and also a great deal in your quads.
The best body boarders are generally smaller body types with incredible flexibility.
And, you are right to a certain extent, that working on just building muscle will reduce your flexibility. I am by no means an expert, but what you want is a combination of strengthening, endurance, and flexibility. The big body builders have very little flexibility.
So, my recommendation is to lift weights, but also add in lots of cardio and stretching (yoga is great for stretching). And, when you lift, go for less weight and greater reps to increase endurance.
For both surfing and bodyboarding, one of the most important areas to focus on is your core.