without seeing what you did on the assignment, I am going to guess on what you may have done to get the issue you state.
When you added the student to take the quiz, did you delete the assign to everyone and just add that student in so that there was only one entry for who to assign the quiz to? I just tested this possibility out in my playground, and If you have assigned a quiz to a class, students have taken it and then you want to reassign the quiz to a student; you can get rid of the assign to everyone entry and just put in the single student name. What you instead want to do is add in another entry for that individual student. The good news is that if you did do the above, you can go back and assign the quiz to everyone and get their quizzes back (above is using classic quizzes - not sure about new quizzes)
What you want to see at the bottom of the quiz page (just viewing the quiz; and once again classic quizzes) is the following: (note you will have 1 student as opposed to the student name like mine - that comes from a javascript that I have running)
If you only have the one line stating for 1 student, then you have done what I described.
If that is not the case, can you provide some more information/screen shots and someone can probably help with the issue.
edit: I just checked with New quizzes, and the assign to area works the same as I described above for Classic Quizzes. You can delete everyone and assign to an individual student after students have taken the quiz. you can then reinstate everyone and view their results.
