What are the examples of mathematicians who were below the average in their student life (say, upto university level but it may be less) but later in life became significant mathematicians.
Up until now, I haven't found any such example. I have read somewhere that S. Ramanujan failed in the college exams but secured highest marks in mathematics. The failure was due to securing pass marks in other non-mathematical subjects.
I think that such examples are hardly possible. The reason behind this speculation is that those who are to be significant mathematicians are those people who are expected to like mathematics. And if they like mathematics then they probably don't tend to ignore mathematics (I am assuming that each such people has at least the ability to pass the exams if they read seriously).
Still, there may be some people who love the subject so much that they don't bother about the progress in their school or college and devote their time to self-learning. In this case, they may tend to neglect the exams and simply go on finding the gems of mathematics, largely by their own or by some the help of some other expert (or experts) in this field.
More of an engineering story, but a Mathematician nonetheless:
Michael Phelps was a boxer on his way to the golden gloves. But he was in a horrible car accident that crushed his skull, this should have ended his boxing career. He tried college, got a job in a girls' sorority house, had a ton of fun, but failed out. After failing out of college, he traveled to Oregon to box. He got knocked out and hurt badly; he woke up on the bench of the locker room. He started to make his way from Oregon back to California.
The trip would take several months. He had no money, and could only be in a car for a short period of time before vomiting. So he would hitch hike for a while, and then try to find a job after the few miles of driving he could manage.
Once back, he confessed to his coach what he had done. His coach took him for a walk and had a talk with him. He said, "It's time to go to school. School is just like boxing. Studying is training and tests are like fights." Michael Phelps enrolled in college again. When he did so, he went to the counselor and asked "What's the hardest major you have?" Math was the answer. He double majored in Math and Chemistry. He then went on to get his Ph.D. in chemistry.
Dr. Phelps is now the chairman of Department of Molecular & Medical Pharmacology. Most notably, Dr. Phelps played a lead role (perhaps he was the inventor) of Positron Emission Tomography. For those who don't know, this is a very common medical imaging technique used to screen for cancer.
Source: He told me this story himself, but you can read a bit about him here: http://en.wikipedia.org/wiki/Michael_E._Phelps