I was always a bigger fan of Luke than of Han. Probably because I was so young when I first came to them, or they first came to me. Luke made for the more interesting fantasy.
So, I'm looking up articles about this, Hand versus Luke, and I really wish there were a serious study, but I might be the only person who would think studying that would be worth it for the sake of, say, a film program, a communication studies program, a psychology program. Glenn Geher, PhD, writing for Psychology Today, 5 March 2013, talks about a class discussion in a graduate course in social psychology...
The conversation touched on several themes relevant to evolutionary psychology--mate choice, optimal features of long-term mates, optimal features of short-term mates, morphological features of sexually attractive mates, the handicap principle applied to high levels of testosterone, inbreeding depression, and so forth...
Let's unpack a bit of that before moving on. Evolutionary psychology is specifically about "useful mental and psychological traits--such as memory, perception, or language--as adaptations, i.e. as the functional products of natural selection" according to Science Daily. That means, in this case, you could get into why Han behaves the way he behaves--old backstory had him as an imperial cadet (I think he was just a cadet) who left the service of the empire when he saved Chewbacca from slavery (under the Empire), hence Chewbacca's life debt to Han (which is not mentioned once in the released films, of course; it's one of those things people just know, like ewoks being called ewoks), and then he became a smuggler, because he had charisma, he had great piloting skills, and he was a gambler... Of course those aspects raise more questions about who he was before he was an imperial; or why Luke behaves the way he behaves--this one is more obvious because the film shows us a snapshot of his life; he's got friends, they fly vehicles called T-16s (the film doesn't tell us this, but the small model ship he's playing with early in this film is a model of a T-16), either racing or hunting womp rats, but his Aunt Beru and Uncle Owen are a little overprotective of him, and while we don't know exactly why they are the way they are, we do know they fear he will turn out like his father, but do they even know that Anakin turned to the dark side, that he's Darth Vader? Did Obi-Wan tell them that when he dropped off the newborn Luke? Also, Owen relies on Luke to help work their moisture farm. Maybe Owen just can't hack it anymore, and Luke leaving (if Beru and Owen weren't murdered by stormtroopers) would have put them out of business. "Well, he'd better have those units on the south ridge repaired by midday or there'll be hell to pay" and all that. Luke is young, he's whiny, but he's got dreams. He dreams of joining the rebellion against the empire.
As for the long-term mates/short-term mates thing, there's this line from that same Psychology Today piece, from a friend of the author after hearing about that class discussion: "[C]an't it be just that Han Solo would be way better in bed? It's not like I'd want to have babies with him!" Or Dave Golder's take, writing for Games Radar, 6 February 2013: "I'd love to go out with Han sometimes, but it's Luke I'd take home." He's paraphrasing Jean Grey from X-Men 2: "Girls flirt with the dangerous guy, they don't bring him home; they marry the good guy."
Golder makes the case for Luke over Han based on 5 (misnumbered) criteria: 1) Luke is an idealist, 2) he has a lightsaber, 3) Han is "kind of a jerk", 5) Luke's a badass, 6) Luke can use the force. If you know that last one is hereditary then that's a great point in his favor when it comes to picking one of these guys as a mate. #5, though--Han is also a badass, just a different kind of badass. Luke is more of, at the same time, a dreamer and a planner. Han just makes shit up as he goes and risks everything; he's a gambler.
Since Luke just did his dramatic double take on his dead aunt and uncle, I gotta mention, a big plus for me in this film is that Mark Hamill is far a better actor than Harrison Ford. If you know the story, Ford wasn't even supposed to be in the movie. He was a carpenter, but they needed someone to read lines and George Lucas liked his delivery. Han has some great lines, and Ford has charisma, but Ford has never been much of an "actor." (He does have some great delivery occasionally, though, like his bit about the force:
Hokey religions and ancient weapons are no match for a good blaster at your side, kid...
Kid, I've flown from one side of this galaxy to the other. I've seen a lot of strange stuff, but I've never seen anything to make me believe there's one all-powerful force controlling... everything.
There's no mystical energy controls my destiny! It's all a lot of simple tricks and nonsense.
Han's atheism might appeal to me now, but as a kid, the idea of the force, and being able to channel it to control people and move objects was awesome.
As for morphological features that make them sexually attractive, I don't think I was sexually attracted to either character when I was a kid, but I must say, looking at them now (looking at them in the movie right now, not looking at Hamill and Ford as they look now), I think Luke's loose (slightly feathered?) mop is more attractive than Han's parted (but not nearly combed) hair and those 70s sideburns... Ugh, now that I noticed them, I can't not see them. They're not long or anything, but they stand out. Anyway, I prefer men more rugged than either of them, like Ewan McGregor in Revenge of the Sith but less clean. But really, my attraction was for Leia, of course. But, I think I'll save talking about Leia for tomorrow.
Back to Psychology Today. The handicap principle is a weird one here. It applies to things like a male peacock's big tail; the male's prowess is demonstrated because he can thrive despite the extra big tail. I'm not sure, in that class discussion, where the "extravagant male trait" was exactly. "High levels of testosterone"? Han's swagger? Like because Ford was 9 years older than Hamill, he's more of a man? His voice is lower, sure. He's taller. And, as far as cinematic visual cues, go, Han is the cowboy, with his low-slung gun belt and his trusty sidekick (Chewbacca) and steed (Millennium Falcon). He's the adult, Luke's the kid. But, this just circles me right back to where I started. I think I liked Luke more because I was younger. That mythological angle, learning that your dead father was a great warrior and you can use the force and take down an evil empire--that's what a kid wants. What I wanted anyway.
(Plus, while Han and Leia would get together, Luke would eventually hook up with Mara Jade and she seemed like a pretty good catch... But, alas, she's no longer canon.)
The films set up plenty for both Luke and Han to do, though. Because, that's cast calculus. Swashbuckler rogue and monk/sorcerer, with Chewbacca the barbarian, Leia as a fighter with noble background, and in this film by itself, Obi-Wan ain't much more than a monk or a paladin. He barely uses the force, and hardly uses his lightsaber. Of course, he dresses like a samurai...
And, I'm drifting.
I wish I had time for a survey. Find the confluence of age at first contact with Star Wars and attraction to Luke or Han. But, I don't.