The lookup table suggestion by PolyGeo is probably the best way to go if there are a lot of different values in the Name field, especially if you don't know what those values may be. Start by creating a Summary Statistics table of the Name field to get a table with all of the unique values in the field. Then add your second field and populate it. Then use the JoinField tool or do a standard join and then export the shapefile layer to create a new shapefile with the field.
If you already know the set of possible values in the Name field and there are around 30 or fewer different values you can use the Field Calculator as userBZ suggested. If there are 7 or fewer known values in the Name field, the If Then ElseIf Else syntax works in VB Script. However, if there are more than 7 different values you must use the Select Case syntax for VB Script. Python has similar If Elif Else syntax and Switch Case syntax structures. The Select Case syntax processes more efficiently, but it is limited to literal value comparisons and not other kinds of logical comparisons like <, >, etc.