If you tick the responses option of 'While the quiz is still open' in the Review Options in the quiz settings, the attempt number in the quiz info will become a link to the quiz review page, for all students though; you cannot limit that option to one particular student. Also, that's not re-opening the quiz but just allowing review. If you want to allow the student to re-attempt and keep number of allowed attempts on one, you will have to delete the student's current attempt. 
Thanks!
How do you delete the student's current attempt?
You go to the list of attempts tick the checkbox next to those attempts you wish to delete and then click the 'delete selected attempts' button.
PERFECT! Thank you SOOOOO much!!