How much does it cost to ground a B737 max aircraft for an airline?
Well we can look at another grounding case to see what it costs.

Boeing grounded the 787 Dreamliner. This grounding lasted about 90 days (at least as of the estimated costs by ANA, I think the actual grounding lasted about 116 days). 

Japanese carrier ANA had the largest 787 fleet at the time with 17 of the 51 in operation estimated it's costs at $15 million. 

Doing some simple math, \$15,000,000 for 17 aircraft works out to be about \$882,352 per aircraft for the 90 day grounding. This is about $9,800 per day/aircraft. 

They also estimated that this resulted in ¥1.4 billion in lost revenue. This is \$12.7m USD (in today's dollars). That gives a lost revenue of about $3320 per seat, per day (averaging 250 seats per aircraft). 

Now taking into account that there are 393 737 MAX aircraft as of the grounding, with an average seating of around 190 passengers, this could cost airlines significantly. Just in terms of costs, extrapolating the 787 numbers and assuming they are grounded for 6 months, looking at about half a billion dollars (*assuming the costs for grounding the 737 are 75% of the 787 in terms of storage and maintenance) for the entire industry. 

In terms of lost passenger revenue, we can assume the numbers are about on par with the 787 since it is a more heavily utilized aircraft (shorter routes, more flights/day). Using the same 787 numbers we would be looking at about $44.6 billion dollars, again, industry wide (and yes, this number seems extremely high to me, but working from what ANA claims). 

Can they claim this cost for Boeing? Maybe. They may be able to recover the costs of storing and maintaining the aircraft during the grounding. It is unlikely that they would be able to recover the lost revenue (and it would put Boeing out of business to do so). We'll have to see when this all washes out what the airlines do with Boeing. Most likely they will negotiate deals with Boeing for preferential pricing on future aircraft or some other kind of cost-offsetting deal.