I have previously (yesterday) been able to save a Spatialite layer from a FileGDB feature class using the Right click/Save as... method in QGIS. Today, however, with no changes to any software (tried on 2 machines), the creation of the Spatialite layer today does not appear to include the OGC_FID field.
The OGC_FID was actually created. I had previously NOT been using the option in the save as dialogue to add the layer to the map, and added it manually afterwards. It seems that enabling the option to add the layer to the map causes the OGC_FID to be hidden. Perhaps this is a bug?