Although they stick to the board for the most part, they will deviate if a coach feels strongly and puts together a case.
Have never been in a room where a legit NFL team was legit building their legit draft board... I'm sure that will surprise everyone reading this post... but having said that...
I've been in a room where something similar was being built, and I've often been made to wonder if there is some reason the Cowboys (and others) would build their draft board any differently.
That is, it makes sense to rank players based on their positions, and then makes sense to rank positions according to their priority. But in both cases, quite often, there are ties... which, translated, means it's more valid to look at the board as a series of tiers/pools of individuals rather than--as so typically appears on the variety of internet-accessible "big boards"--an outright 1, 2, 3, 4.... 298, 299, 300 ranking of individuals.
So, this conception that one player is the absolute only player that should be an option if one is "choosing strictly by their board" would, given that concept of a board, be highly unlikely.
Further warranting my suspicion that I'm not terribly wrong to presume that is the fact that we all heard Jerry confess that they were deliberating about not two... not three... but
six different options at #99.
Seems most likely, based on that comment, that their quantitatively-organized "board" looks much more like a set of tiers than it is some 1-2-3 listing that is commonly imagined... and thus, that's what would naturally allow, if not invite, some degree of substantive discussion as the draft evolves.