Thanks, guys! I thought that abilities is the noun that is influenced by the article. Student's is possessive and thus I considered it as if it were an adjective or nominative noun. You wouldn't use a with something like measure of his abilities or measure of physical abilities or measure of survival abilities(ok, the last one might sound awkward but I can't come up with anything that would be closer to nominative noun off the top of my head), would you?
We would, in fact, say "an accurate measure of the abilities of a student." When "a student" gets genitivized, it stands in for the definite article, since we're expressing the definitiness of the abilities through another means (tying them solidly to one student).
Perhaps the confusion is about how "abilities" is plural, in the first place. The point is that any one student has multiple abilities, so "abilities" works as a name for a set (singular) of abilities (plural). That's why we use the definite article in this sentence.
It's possible to use the article a/an with "ability," but not with "abilities."
Numeracy is an abilityof every student who graduates from this school.
Numeracy is a necessary abilityfor a student who wants to graduate.
A written examination is not necessarily a true measure of anability as a musician.