Adding value to column in a filtered table and maintaining filter button

SarahBaizer 

The error you're getting has nothing to do with Filter or the Filter Button.

It's a matter of Data Validation; from the appearance of the second screen shot, you're trying to enter a name when what you need to be doing is selecting "Not Yet Assigned" and then (presumably) adding the new name to a table somewhere else in the workbook.

There is a list somewhere of the names that are deemed acceptable. To find that list you'll need to place your cursor in one of the cells in that column and then enter the Data Validation dialog box (accessible from the Data toolbar or from the Data menu at the top of the screen)....and see what turns up; you'll either see a list IN the Data Validation dialog box OR there'll be a reference to a sheet and set of cells that will contain the current list.