Trillian is this amazing… oh, who are we kidding? Arthur gives Trillian a terrific description, noting that she's "Beautiful, charming, devastatingly intelligent" (13.63).
So far so good, but we have to say that Trillian doesn't get to show off those qualities very often. She may be beautiful and charming and intelligent—certainly, she gets the attention of both Arthur and Zaphod at that party in Islington. She definitely seems like she's supposed to be a love interest for Arthur: it's easy to imagine the Hollywood version where lovable loser Arthur accesses his inner action hero, saves the world, and ends up winning the love of Trillian. In fact, we don't have to imagine it: that's kind of the way the movie version does go. Knock yourself out, if that's your sort of thing.
But here, in the book, Trillian doesn't really get a lot to do. That's a real shame, because she does show off two very important qualities in those few moments when she gets some of the spotlight. (Admittedly, it's hard to get the spotlight away from Zaphod.) First, she's smart and very math-friendly; second, she's what we'll call "adult-nice."
Trillian might be the smartest of the four organic main characters (Arthur, Ford, and Zaphod being the others), especially when it comes to math. As she notes to Arthur, she has a degree in math and astrophysics, but there's not a lot she can do with that on Earth. She left Earth with Zaphod because "with a degree in math and another in astrophysics what else was there to do? It was either that or the dole queue again on Monday" (13.69). Translation: dole = welfare; queue = line.
What all that means is that Trillian's math and astrophysics degrees couldn't get her a job. Now, that's a pretty clear indication of how screwed up Earth is: we have lots of jobs for bulldozer drivers to destroy houses, but no jobs for math whizzes who might be able to find the alien fleet before they destroy the Earth.
So it's often up to Trillian to deal with probability, like when she points out to Zaphod what improbable coincidences they are encountering, like picking up two guys from the same area where she's from (12.17-38). She's also more responsible than Zaphod, protecting him from himself. When he's gesturing excessively with worry, "Trillian quietly moved his hand before he tapped anything important" (11.9). This nicely shows that Trillian knows what controls are important on the Heart of Gold spaceship, even though she's just an Earth-person and not the president of the Galaxy who stole that very ship.
Also, we don't want to harp on this, but it is pretty cool that, in 1979, Douglas Adams seemed to think that women could do math and figure out spaceships.
Being smart is great and all, but it's more important to be nice, right? Most of the time, Trillian is the grounded human being who tries to prevent craziness. For instance, she remains calm when Zaphod freaks out about the hitchhikers: according to Trillian, they are "Just a couple of guys we seem to have picked up in open space" (11.7). To us, that use of "just" and the casual word "guys" seems to have the subtext of: "Simmer down, Zaphod. I'm on this."
In fact, whenever she talks to other characters, Trillian is constantly keeping cool. Her speech tags are often marked by "patiently," "calmly," or other adjectives to show that she's not losing her mind. For instance, when she needs to explain something to Zaphod, we get a glimpse inside her brain that shows us that she's inwardly very annoyed at how stupid Zaphod can be: "Parts of the inside of her head screamed at other parts of the inside of her head." And yet, when she has to explain the sitch to Zaphod out loud, we get this: "She said, very calmly…" (12.31).
We can see the same thing when Trillian talks to the other characters. When she talks to Marvin, she's totally nice: she speaks in "a bright compassionate tone" (11.32) and in a sing-songy voice—she lilts and lilts and lilts (11.47, 49, 51). And yet, when Marvin leaves, Trillian's lilt vanishes: "'I don't think I can stand that robot much longer, Zaphod,' growled Trillian" (11.55). So we see that Trillian is a two-faced liar, right?
Well, sure, she's not letting her real feelings show, and she probably has terrible stress because of that. But at the same time, we can see that Trillian is the adult peace-maker here. While Marvin wants to bring everyone down (on purpose) and Zaphod is trying to blow everything up (accidentally), Trillian is the one who keeps everything moving safely along. She's the only responsible adult in this group. It's no wonder that she doesn't get a lot of the spotlight: as the only competent character in the book, there's not much that's funny about her.
Truth be told, Trillian doesn't have a huge role in the book. Zaphod causes problems, Arthur suffers from the problems, and Ford explains the problems. And Trillian… well, she kind of keeps the peace and prevents Zaphod from killing everyone. But she can't fix all the problems that Zaphod causes or there wouldn't be much of a story.
We first hear the name Trillian in Chapter 4, where she's described as "humanoid" (4.45). So it's not clear that she's just another human until Arthur sees her on the Heart of Gold in Chapter 13 and identifies her as "Tricia McMillian" (13.68). Hold your hand up if you were surprised by that. Adams sets it up so that we're expecting a bona fide alien, but she's really just the girl next door.
Let's note, though, that boring old Arthur Dent doesn't start going by the space name of Ardent or something. Arthur is usually confused by the things he finds in space because he still has the mindset of someone from Earth. By comparison, Trillian's name shows that she's more comfortable in the weirdness of space. She's willing to change and adapt, whereas Arthur mostly wants tea.