The basic story is thus: 17-year-old Ataru Moroboshii is an inveterate lecher. He never saw a pretty girl he didn't like, and his life is an endless series of attempts to get pretty women to go out with him. At the beginning of the series he is in a steady relationship with Shinobu Miyake.
He's out for a walk one afternoon when he is abducted by a mysterious government man and forcibly taken home, where he sees that his parents have a guest: Mr. Invader, an Oni. Invader tells Ataru that he has been randomly chosen to take part in a contest against the champion of the Oni. If Ataru wins, Earth is safe; but if he loses, Earth will be invaded and subjugated by the Oni.
Ataru refuses--Mr. Invader is huge--until he learns that his opponent is to be the lovely and curvaceous Lum:
...and of course he immediately agrees to take part in the contest. (By God, what man wouldn't??)
The contest is a simple game of tag: all Ataru need do in order to win is to grab Lum's horns; and more, he is given ten days to accomplish this goal. But on the first day of the contest he learns, to his chagrin, that Lum can fly....
The night before the last day, Shinobu tells him that if he wins the contest, she'll marry him. With this motivation, Ataru finally manages to come up with a winning strategy...and while exulting in his victory he says, "Now I can get married!"
Lum misunderstands him and says, "All right! I'll marry you!" And the worldwide press jumps onto the bandwagon, calling it a "goodwill marriage". Ataru's fate is sealed.
Now for the musings.
What follows is entirely my own theories, based on what I've seen of UY. You may have other ideas.
Lum and Ataru are of an age--around 17. Lum is (naturally) very pretty, and looks nothing like her father.
An "Oni" is a Japanese ogre. In UY, Lum and her race are aliens, and in one episode she differentiates between her race and the more traditional Oni from Japanese folklore: "We're science fiction! You're fairy tale!" Even so, many of the Japanese customs regarding Oni seem to work on Lum and her cousin, Jariten.
Fairly early in the series we learn than Lum had a fiancee on her homeworld, an Oni man named Rei. Rei is spectacularly stupid and turns into a giant "cow" when agitated; he is also very handsome, and Lum can't stand him.
Finally, Ataru is referred to by many people in the series (primarily men) as "ugly".
Here's my theory:
Lum is slated to marry a man she can't stand, Rei. She has no other prospects, because on her homeworld she is considered ugly. To us, she is beautiful; and her fiancee Rei is equally handsome to the women in the story. But Rei is also--to Oni--hideously ugly.
To help her improve her prospects, her father decides to allow her to act as the champion of her race in the pacification of a moderately advanced technological civilization (Earth). According to Oni customs a planet may not be invaded unless it's already at war with the Oni, so they must be given an opportunity to escape invasion. They deliberately choose a world on which the indigenous species (us) has no special powers such as flight, energy attacks, etc.
If she is successful in the contest, Earth is subjugated by the Oni, and the prestige and/or wealth from that conquest makes her desirable enough that her physical attributes are less important. If she is not successful, she has lost nothing.
Their computer "randomly" chooses a champion for Earth who has little athletic ability and who is essentially a bumbling, incompetent, lecherous idiot with bad luck. Invader successfully "sells" him on accepting their offer, inversely, by first making him think that Mr. Invader will be his opponent, and then showing him Lum.
But Ataru surprises them: he wins at the last minute. And here his "bad luck" comes into play; his victorious exultation gives Lum an opportunity.
He mentions that now he can get married, and Lum purposely misinterprets what he says as a proposal and accepts it, telling him that marriage proposals are considered sacred on her planet. Thus she escapes having to marry Rei, who (to her) is hideously ugly, and instead is able to marry a man with average looks.
The issue of fame and fortune becomes a non-issue since she has accomplished her primary goal of marrying someone she can stand to be around. Ataru is not the brightest star in the sky but he's a genius compared to Rei; and his flirtatiousness is his worst attribute--certainly that's better than turning into a cow. For Lum, Ataru is a serious step up.
I say "bad luck" above because Ataru later realizes that he doesn't actually want to be married to any woman. His sudden marriage to Lum (in Japan, marriage requires only the appropriate entries made in government records) has robbed him of his youth: he can no longer "play the field" but is tied to one woman. In episode 10 he admits that he loves Lum, but he also loves Shinobu; Lum's friend Oyuki; the gorgeous school nurse and Shinto priestess Sakura; and several other attractive women who have shown up in the series since episode 1.
Lum falls madly in love with Ataru, regardless of how he treats her. She gets her licks in, though, since she has an electric attack which is apparently powerful enough to affect the power grid, and she has the temper of an ogre (or "oni"). Considering what his personality is like, his looks must play a factor.
* * *
Overall, UY is a fun series, although for me it was an acquired taste; the first time I saw it, I hated it. Well, the story improves vastly once you get past the first handful of episodes, and Lum is the ultimate in anime babe hotness, anyway. I don't know what value, if any, this theorizing has, but it's fun to think about.