Relationships: Snow
"Going out of his way to help someone? That's Snow all over."
Even before the game was released, the developers often mentioned that Lightning and Snow have a unique relationship. Their connection through Serah is definitely not generic. Further more, it's their personality differences that mke their relationship so interesting. They are superficially as opposite as fire and ice. Light is cool, serious, and quietly introspective. Snow is hotblooded and action-oriented, just being around him either tires you out or pumps you up. Let's look at how their dynamic connection matures through the game.
A Threat to Family
"He's too stubborn to die. And that's his best quality. He's arrogant and chummy from the get-go. He thinks he's everybody's pal. Never liked him much. He leads a bunch of kids--gang called 'NORA.'"
I've already established several times that Lightning's relationship to her sister Serah defines and motivates Light. Thus, Lightning sees Snow as a threat to her family in that he's taking Serah away, leaving Lightning alone. Lightning and Snow first met on Day Two, according to the FFXIII Official Guide. Naturally, because Lightning is part of the military, she would disapprove of Snow's NORA vigilantes, seeing them as dangerous and irresponsible. Next time they meet, Serah is cursed & engaged. It didn't do much to counter her first impression. She now associates him with all these changes to Serah that make Lightning feel like she's losing control of her family.
Not So Different
When Lightning first meets Fang, the latter reveals how Fang and Vanille indirectly contributed to Serah be chosen. Lightning punches her. Fang then says, "You sound exactly like Snow. And he didn't hit me." WOW, you can see Light flinch after hearing that. That simple line pointed out two things to lightning: (1) Snow has similar values as Light, despite doing things differently. (2) Not only is Snow similar to her, he's actually BETTER than her. He didn't childishly take out his anger physically on Fang.
"You've been a total idiot. But still...It saved me.
You trusted Serah. Let that faith drive you. It even made me want to believe."
The apology Lightning gives Snow at the Estheim Residence is the turning point in their relationship. She vaguely asks him to forgive her for everything. Basically, for being an elitist jerk during their first meeting, for not believing him on her birthday, for not recognizing how he cared for her sister, and for punching him several times. Afterwards, we see Lightning being nicer to Snow, such as complimenting his kindness when he argues to help out the Cieth Stones or lightly teasing him for his tactics.
Trustworthy Confidante
Snow: "We will see her, right?"
Lightning: "Don't do that. There's no room for doubt.
We'll see her again and soon. You convinced me of that. So stay strong."
On Gran Pulse, after spending much traveling the planet and fighting together, we see that they have become confidantes of sorts. In the cutscene among a field of dandelions, they talk to eachother about Serah. Though it's hard to tell with Light, she implies that she approves of Snow and Serah's marriage. She also puts her fist to Snow's back when he doubts the future, telling him to stay strong. As she does this, Lightning sheds silent tears that show she has the same fears.
Interestingly enough, not only has Light finally accepted Snow as a friend, they've also switched places. Before, Lightning was the negative personality who eventually becme inspired by Snow's relentless faith. Now, as Snow's concerns surface, Lightning is there to reinforce his previous optimism. This dandelion scene on Gran Pulse is really touching and significant. It shows the vulnerability of both heroes and how they emotionally connect to find support.
As a Pairing
I adore this pairing! It's the classic Opposites Attract dynamic. In pairings of opposites, the characters always clash in way that are funny, passionate, or all of the above. I love it. My reasons for liking this pair: (1) Their relationship development in the game. A relationship is more notable when you earn it through growth and appreciation. (2) They look good together. Lightning does not look like a little kid (sorry, Serah!), and Snow does not look too old for her. (3) Once again, the Opposites Attract chemistry. It's a fun and complimentary pairing that's sure to bring laughs.
I recognize that this shipping is a creation of fandom, and only popular with a small portion of fans too. This is because of the Serah Factor. Snow x Serah is completely canon. Snow loves her loyally and intensely. With the dedication he shows in the main storyline, it's hard to imagine Snow with anyone. So for SnowxLight to work in fanfiction, rp communites, etc, fans need to properly and creatively address the Serah issue. I could certainly see a realistic outcome where Snow and Serah decide they were too young, or their relationship was too rushed, and Snow & Lightning grow close due to work/missions they embark on with each other. But for the most part, I admit it is not the most probable pairing.