Related to answer I gave to a similar question ([determine min and max elevation ... within my current extent][1]), I wonder if this would work:

    import arcpy

    # this sets extent to current display, you can instead set it to ROI polygon
    arcpy.env.extent = arcpy.mapping.MapDocument.activeView.Extent
    
    # for a multi-band raster, pay attention to the band index (last input)
    arcpy.MakeRasterLayer_management("c:/temp/raster.tif", "rlayer", "#", arcpy.env.extent, "1")
    
    # Get the geoprocessing result object
    bandMinResult = arcpy.GetRasterProperties_management("rlayer", "MINIMUM")
    # Get the elevation minimum value from geoprocessing result object
    bandMinDN = elevMinResult.getOutput(0)


  [1]: https://gis.stackexchange.com/questions/107460/using-arcpy-how-can-i-determine-min-and-max-elevation-from-a-raster-dem-layer-w