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How can I fairly compensate a student who showed passion and dedication for my undergraduate course but performed poorly on the final exam, without unfairly advantaging them over other students?
Additional context: The student spent a significant amount of time taking meticulous notes for the course and shared them with others, but this was not announced as an opportunity for extra credit beforehand.
With respect to considerations of this kind one should reflect on one's information sources. For every student who exhibits to the professor hard work there are other students who have worked just as hard but have not exhibited it to the professor. If exhibiting activity to the professor does not form part of formal evaluation procedures, it should not be taken into account, as this is certainly unfair to other students.
If one desires to reward effort rather than accomplishment (perhaps a bad idea!) then one should build this into the evaluation scheme from the beginning. If one wishes to allow for some part of the evaluation to be discretional, based on the professor's subjective impressions (perhaps a bad idea!), one should build this into the evaluation scheme—provided one's institutional context even permits it.
Sometimes one has a situation like the following that often occurs in Spain. Students are graded on a 0-10 scale that steps by tenths. A passing grade is 5.0. The difference between a student who obtains a 4.8 and a 5.0 is often essentially marginal - one could say within the margin of error of the not so robust evaluation process - and the professor decides to pass everyone who obtained at least a 4.7. One possibility is to add .3 to every grade that it at least 4.7. Of course this generates a new threshhold effect at 4.6 ... and so such a procedure can reasonably be questioned. But anyone reasonable will agree that it is less questionable than giving 5.0 to both the student who obtained 4.7 and the student who obtained 5.0. What seems to me wholly unfair is to give 5.0 to some student who obtained 4.7 because they worked hard, while not doing the same for some student who got 4.8 and never came to office hours and appeared (!) to be a lazy bum.
In short, I think best practice is to stick to the previously published criteria. This has its shortcomings, but they are less significant than the shortcomings of the alternatives.
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.
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):
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).
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.
There is a general principle that if you offer any additional opportunity, then it has to be opened to everybody. You can declare pretty much any grading policy in the beginning of the class. My favorite extra options are extracurricular projects, extra hard problems to solve at home and reading assignments (I'm teaching mathematics). There have been cases when people using those options ended up with having enough points for an A even without getting any credit for the finals and I was completely content with that.
What I consider somewhat unethical is offering special opportunities to those who failed miserably a posteriori without giving an opportunity to those who passed without flying colors to improve their grade too, so I wouldn't change the grade itself. However you can always give some extra encouragement ranging from a few kind words to a formal recommendation addressed to a colleague or registrar's office for the student to enter an advanced course for which yours is a prerequisite even if the official grade is below the official requirement (if you think it is appropriate, of course). You can also award some funny prize for good notes or something else. Once my official requirement for an A in a graduate course was solving 5 problems at the blackboard and one student solved 10, so I gave him an official grade of A, but presented him an AA battery on the last day of the class. You can easily invent something like that too: the cost is next to nothing but it is a clear sign of appreciation, and, I think, that is all the situation calls for.
The ideal we strive for as a society is equality of opportunity. We must not mandate equality of outcome - the classic satire Harrison Bergeron is the logical conclusion of that.
By all means attempt to make your assessment procedures as fair as possible. Exams may not be the best way to do a fair assessment, so for sure you can look at that structurally. But after that, you must not bias the results. And to prevent you biasing the results in favour of your favourites, you should have a marking scheme to stop you doing it.
At most, you could look at the appeals procedure in case this student has had some kind of life event which has affected their results. But apart from that, you have to let it stand.
Artificially boosting the grades of your favourites is literally on the list of gross misconduct for pretty much any academic job. It's how male teachers reward their favourite female students in exchange for "favours". (Or sometimes in different orientations, sure, but that's the most common one.) You could be fired on the spot and never work again. Seriously, don't go there.
