According to the accepted repository of all knowledge and wisdom, Wikipedia, the Garzweiler II mine contains about 1,3×10⁹ tons of lignite. Taking optimistic estimates, this corresponds to 1,2×10⁹ tons of CO₂. Over the planned 24 years of operations between 2006 and 2030, this represents about a thousandth of current worldwide CO₂ production.
It is estimated (Deltacommissie (2008). Samen werken met water: Bevindingen van de Deltacommissie.) that only the Dutch sea and river defences will require €38 billion in the coming years to adapt to changing climate. World-wide the costs are closer to $100 billion per year between 2010 and 2050 (PBL (2014). Costs and benefits of cimate change adaptation and mitigation: An assessment on different regional scales.).
So the Dutch government should sue the Garzweiler mine for €40 million in climate damages and convince other governments to sue them for the €4 billion in total damages this mine will cost in CO₂ exhaust alone.
I should note that I made a number of estimates that are very much in favour of the coal mine, and in actuality the damages are likely much worse.
Namely, I estimated the mine's CO₂ production by multiplying the lower bound of carbon content in lignite with the mine's estimated production. This discounts all other greenhouse emission effects from destroying vegetation and digging up in a huge area of land, transport of the coal etc.
Similarly, the trend of worldwide CO₂ production is decreasing and in fact the numbers on climate adaptation include investments in limiting CO₂ exhaust. So the 1/1000th of worldwide CO₂ production is very much a lower bound of the actual proportion of damages.