If I was that guy and I just wanted to go down swinging, I would make the following arguments:
25/162 is the only person with the number 15 in their percentage rank. 24/162 is 14.blahblahblah and 26/162 is 16.04blahblahblah.
So, 25/162 would be the only person impacted by a lax interpretation of 15% (so to forestall slippery slope arguments)
With that in mind, "Dean's List" is an honor that makes it easier to get jobs. So it is a functional tool. Job placement matters as much to the school as it does to your friend. One extra person will not dilute Dean's List. But, it will make one easier job placement, which will benefit the school. (utility and mutual benefit: Getting to Yes for the win.)
Then, your real only argument for why your interpretation of 15% should win goes as follows.
"The handbook is vague. 15% could include everything up to 16%, so 15.9999% and under. Your book doesn't state either way. Because both student and school benefit more under my analysis, and the vagueness gives a justification for the grant of Dean's List, you should do it."
Then I'd make a meta-argument. "The fact that I'm here making this argument shows you that I'm dedicated and a hard worker. It shows you that I am thinking about my job prospects and how it helps this school." The theory being that the school will like an aware dedicated individual who has the school's interest in mind.
The problem is your friend probably has no leverage, and the schools main concern will probably be about not diluting Dean's List, and not setting a precedent of case by case determination.
The only leverage I can think of is, "My rank makes it possible to transfer to a school that will make it easier for me to get a better job. Dean's list would help me get a better job if I stayed here. My only goal is to get a better job. I would prefer to stay here and get a better job, but if I cannot get a good job here I will leave."
I'm interested in hearing how this turns out.
I think your friend loses, and I think your friend should lose. 15% is a number. Percentiles are well understood. I don't think there is any vagueness that actually exists. He missed the mark just slightly, and that sucks. But someone has to be the first person to not get the grade. Like the first guy to not grade onto law review.