What methods can I use to create a fishnet from thematic polygon data

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I need to create a regular grid containing attributes from other thematic data from ecological data. I am in effect creating a homogenous polygon raster and adding information from another dataset to the raster cells which contains irregular-shaped polygons. I need to therefore clip or union the fishnet with the thematic data - but the only problem is that inside the fishnet cell only one value is permitted. How do I decide which value the cell obtains - some features are more important than others and should be weighted, other cells would just obtain the value from the polygon with the largest area inside the cell.

Below is an illustration of the data in ArcGIS.





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