I'm not a scholar of these things but I thought it was an interesting question, so here's what I was able to find. Most of the following particularly concerns the mode of examination at Cambridge, since that seems both to be the earliest example of formalized grading and information is less plentiful about other universities:
That far back, there was no formalized grading system. Evaluation of students was informal, and to obtain a degree one would pass a series of oral examinations, which seem to have functioned on a basis of pass/fail with the prestige attached to a degree more relating to the reputation of one's professors/tutors than to a formal result. At Cambridge, those who obtained degrees were listed in the Ordo Senioritatis; it seems that this did not become an order of merit until the 18th century.
Often credited as being the origin of formalized testing in higher education is the Cambridge Mathematical Tripos. From 1710 we see a division between 1st and 2nd Tripos; from 1747, the result lists were printed and published, ordered into classes based on merit: Wranglers (of whom the best-performing student was the Senior Wrangler, the second-best Second Wrangler, both titles conferring significant rewards both in prestige and economically), senior and junior optimes. At first these were oral examinations. Other universities would soon follow in their footsteps as examinations and grading became more formalized in the 18th and early 19th century. The first record of university "grading" in America comes from the diary of then-president of Yale University, Ezra Stiles, who in 1785 wrote in his diary of four grades of graduates: Optimi, Second Optimi, Inferiori and Perjores.
Interestingly, in the 16th and 17th centuries, even well into the 18th century, any master at the university (at least at Cambridge) had a right to question candidates for degrees, and often took an active role in examinations, without necessarily having taught the student in question or belonging to any sort of formal examination comittee. This was a right enshrined in Elizabethan-era statutes (Elizabeth ruled from 1558-1603), which was not formally abolished until the early 19th century.
This led to problems as faculty members of the various colleges would show blatant favoritism towards their own students, and academic rivalries between the colleges meant that faculty from one college would show up to mercilessly grill students of rival colleges in order to further their own standing. This, in turn, led to a change where students would not be examined together with their fellows from their college, but were instead divided into groups based on perceived ability. This change was instituted by the moderator of the Senate House Examination (i.e. the Tripos exams) in 1756, Richard Watson, who commented that, "by this arrangement, persons of nearly equal merits are examined in the presence of each other, and flagrant acts of partiality cannot take place."
Although the Senate House Examination was supposed to include some moral philosophy, its chief focus was mathematics, thus giving the exam the name the mathematical tripos (the term tripos is obscure but may reference three-legged stools on which candidates sat when examined). John Gascoigne quotes a student of Classics in 1748 complaining about having to abandon his literature and language studies to focus on math for the exam, since "one's Reputation in this place depends in a great measure upon the progress made in [natural] Philosophical learning." William Heberden, physician and Cambridge fellow, complained in 1792 that "... the public honours of the University ... are distributes merely according to mathematical merit, unless one evening dedicated to an examination in morality, to which no attention is paid in ranking the candidates, may be called an exception."
It seems the formalization of examinations in English universities, with a meritocratic ordering was an attempt by universities to assert authority, as they feared government oversight would take away their right to teach and organize themselves as they saw fit. Then, as in more recent times, scholarly institutions were associated with contrarian and politically explosive thinking, and Oxford in particular was associated with the Jacobites, who sought to restore the throne to the Stuarts. This created pressure for more government oversight of universities, and the formalization of the Tripos can be seen as an attempt to assert ownership over examination preemptively--just concerns over properly ranking students on merit were not the sole reason. But they played a part, of course: there was a familiar gripe about the erosion of standards, as some oral examinations were little more than the parroting of memorized material almost verbatim by students who were in some cases illiterate.
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