You've already found the "right" way to do this (digitize the lines, keeping track of a unique ID to join back to the attributes), but here is a spectacularly wrong way to do it.
1.) Make a note of how many records are in your table.
2.) If you table isn't in DBF format, make it so.
3.) Create a new line shapefile (mine is called "bad_idea.shp")
4.) We need to make as many null geometries as there are records in your table. You can do so with the following Python code:
>>> insCursor = arcpy.da.InsertCursor("bad_idea","SHAPE@")
... for i in range(7):# change to the number of null geometries you want to make
... myArray = arcpy.Array()
... myPoint = arcpy.Point()
... myArray.add(myPoint)
... myLine = arcpy.Polyline(myArray)
... insCursor.insertRow([myLine])5.) Close down ArcGIS to release any locks on bad_idea.shp
6.) Delete "bad_idea.dbf" in file explorer
7.) Rename your original table (the one with the attributes you want to keep) to "bad_idea.dbf"
8.) ArcGIS won't open "bad_idea.shp" anymore (I'm not exactly sure why), but QGIS is more liberal with that sort of thing. Bring this diseased shapefile into QGIS and save a copy, say "bad_idea_copy.shp"
9.) Bring "bad_idea_copy.shp" into ArcGIS. It should have your original attribute table associated with null polyline geometries.
10.) Start editing.
11.) Select the feature from the attribute table for which you want to update the existing null geometry to a real geometry
12.) Start editing vertices (editing toolbar)
13.) From the edit vertices toolbar, select Continue Feature Tool
14.) Draw your new feature. This geometry is associated with the selected record in the attribute table.
15.) Repeat.
Have fun. I take no responsibility for any damage this method may cause.
