I found what I believe to be the original source for the theory: A Suggestion as to the Source of Montaigne's Title: "Essais", which was published in 1936. Unfortunately, (for most of you) it's behind a paywall. I'm paying a lot of money towards tuition right now, so I have access and I'll provide some quotes.
It's true that the Floral Games had a tiebreaker known as an Essay:
The Essay was a regulation formulated in 1540 to meet the difficulty which arose of having to choose from among a number of contestants of equal skill and merit. After unanimously awarding the Violet to Jehan Corrière, a student in the University of Toulouse, the judges had difficulty in awarding the Marigold and Églantine, for there were four other candidates whose poems appeared to be of equal value. After deliberation it was decided to put the four to a test (examen). To each was assigned a certain refrain upon which he was required to construct an impromptu huitain or dixain.
It bases the theory off of the following:
It is conjectured from the character of his learning displayed in the Essais that he then took the two years of the course in philosophy at Bordeaux, and that between 1548 and 1554, when he became a conseiller in the Cours des Aides at Perigueux, he completed the law course at Toulouse.
In the sixteenth century most of the contestants at the annual meetings of the Floral Games were students in the University, many of them law students.
Since the connection between the students and the Floral Games was so close, it is quite certain that Montaigne would have been familiar with the annual programs and the trial by Essay. Since the Essay was an impromptu trial or test it is possible that Montaigne caught the idea of his title from it, as he uses the word in precisely the same sense as it was used at Toulouse. Also there is a similarity between the composition of the Essays and the composition of the poems for the Essay at Toulouse. In his earlier essays Montaigne uses an anecdote or a quotation from the ancients as his starting point, around which he clusters his own observations.
As you can see, there's no hard proof that there's a connection. I believe the accepted theory is correct (it's from the Old French essai meaning trial).