Follow your syllabus
Presumably, you gave every student a syllabus at the beginning of the course, and this syllabus outlined standards for grading. The grades you assign at the end of the term should adhere to whatever standard you set at the beginning of the term. To significantly vary from this scheme is unfair to everyone in the class, and, in some places, could lead to your grades being overturned.
If your syllabus says that students need to do well on the final exam in order to pass, then it sounds like this student did not earn the grade you want to give them, per the terms of the syllabus. As such, to give them an inflated grade because they worked hard is unfair to the other students in the class.
Don't do it.
What do your grades mean?
That being said, this seems like a chance to reevaluate your grading scheme for the future. Think about the following questions (among others, but, based on the information you've given, these are the most salient points):
- What should a grade mean? What information am I conveying if I give a student an A? or a C?
- Is the effort that students put into a class one of the things which should be captured in the grade at the end of the term?
- Is the final exam something which should be so central to the way in which grades are computed?
You should then construct a grading scheme which reflects these values.
For example, my belief is that the grades I give in my precalculus classes reflect two things, in almost equal proportion: an understanding of the concepts being taught in the class, and the academic skills which are likely to lead to success in later classes (both mathematical and non-mathematical). I do believe that effort is one of those academic skills, hence I would like to see effort incorporated into final grades. I also believe that being able to accurately complete work in under the pressure of a clock is important to demonstrating mastery of the material (the time limit is a rough proxy for how "internalized" ideas have become).
As such, I have components in my grading scheme which reward effort (students must turn in a certain amount of "drill-and-kill" work, they must participate in class or office hours, etc), and I have components which reward demonstrations of competence (there are graded written assignments, weekly quizzes, and a final exam).
Moral of the story
Don't make one-off exceptions for individual students. In any given term, stick to the terms of your syllabus as best as possible. However, take difficult situations (like the one outlined in the question) and use them as opportunities to evaluate your assessment schemata, and update your syllabus to reflect what you have learned.