Need to Delete "People"

There are several different ways that duplicate names can find their way into the list.  So before deleting any of the People tags, I recommend that you use the Keyword panel to check whether any of the duplicate name tags have in fact been applied to any photos.  All of the Ruth/ruth tags shown in your screenshot should appear somewhere under the People tags Categories in the Keyword panel.  You should click each box separately to see whether any photos show up for that particular tag.  If they do, tag all of those photos with the correct Ruth People tag.  You can then right-click on the duplicate tag and you will have the choice to delete the tag.

Any deleted tags should no longer be offered in the dropdown lists when naming People.