Jana,
If I understand you correctly, the Process GEOGRID File tool worked fine when all other settings and inputs were identical except for the "Regridding Factor"?
I would suggest that you check and make sure that Spatial Analyst extension is on and activated. Next, see if Background 64-bit Geoprocessing option is installed and turned on. This will dramatically increase the memory throughput because it will use 64-bit Python. Your image looks like ArcMap or ArcCatalog. Can you upgrade to ArcGIS Pro? This will also utilize 64-bit architecture and reduce Python-based MemoryErrors.
You may also need to check your output location and make sure to delete any "scratchdir" directories from previous failed runs. Watch your amount of free disk space, and your memory consumption. These could all be reasons for the failure when you move to higher resolutions. Many temporary outputs are written to disk, and the file size stored in memory grows tremendously with every increment of increased regridding factor. Try to use regridding factors that will evenly divide your LSM resolution (e.g. 2, 4, 5, 8, 10, 20).
Let us know how the troubleshooting it goes,
Kevin