SharePoint 2013 Powershell How To Update Modify Lookup Column Field References
When you delete a list referenced from a lookup field, the field stops working, even if you recreate the list with the same name. This is due to the fact that the lookup field referneces the list by ID and not by Title.
Even if from UI interface you can't edit a column definition of a lookup field, you can easily change the lookup definition using Powershell.
The following snippet shows how to change the lookup list and display value of a site column definition. The trick is to work on the XMLSchema of the field and then update the field definition. Note: if you have defined in your lookup field to show additional fields, you have to reset the lookup list for these fields too.