Finding the minimum value:
This may be difficult to do as shapefiles don't fully support subqueries (See link here). However, if the data were stored within a Personal Geodatabase (MDB) this could be done using the following query:
Elev_Diff in (select MIN(Elev_Diff) from <featureclass>)
Exporting a file with the name of its input:
I would suggest adding another tool to the end of the model, Copy Features, to copy the selected features to a new file.