There are a couple of things here. First, the small puddle under your car is more than likely just water coming off of the A/C condenser. I speculate this is what it is due to there not being any smell to the liquid puddle. This is completely normal.
On the flip side though, when the mechanic said you "need a new thermostat" because the housing was leaking, he is more than likely correct (as long as there is actually a leak). Here is an image of what your thermostat housing looks like:
If you look at the very top hole, you'll see the brass looking thing in there. That's the thermostat. It is completely contained inside the housing and comes as a unit. One cannot be changed without changing the other. If the housing is leaking, both would get changed. Due to the composite nature of the housing, it is not unheard of for it to start leaking. As for the $400 price tag, it seems reasonable considering the thermostat in and of itself is over $100. On the Cooper S, everything on the engine is harder to get to, so it does incur some work to get it done. The price would imply about 2 to 2.5 book hours of labor, which sounds about right (to my guestimation).
Obviously I cannot tell you if yours is actually leaking, but I would bet odds on there is probably an issue there. You may want to take it somewhere else and get a 2nd opinion, but it does seem reasonable and feasible.