If its just names you're adding, you could try setting up the form text input name as an array. Let's say you have a text input box to get 1 name. Name it "personname[]" (or whatever you like, just include the "[]" on the end). Have your JavaScript add additional text boxes and keep the exact same name with the "[]" on the end.
Upon submitting, your PHP code will treat $GET['personname'] as an array (try count($GET['personname']) which should give you the number of names provided. Then to access the first one, you could do $_GET['personname'][0] (swap the 0 with the row of the name you wish to retrieve).
This works really well with check boxes and the such and I think it would be a good fit for what you're doing.