Bifrost
While a number of Valkyrie Profile's references to Norse mythology are inaccurate, Bifrost is not. In Norse mythology, Bifrost is the rainbow bridge between Midgard, the mortal realm, and Asgard, home of the gods. Bifrost is what connects these worlds with Yggdrasil, the World Tree. There is also reference to Odin hanging there.
I know that I hung on a windy tree
nine long nights,
wounded with a spear, dedicated to Odin,
myself to myself,
on that tree of which no man knows
from where its roots run.- from Hávamál
Mystina: Well then, I'll go first... What I want is information about Bifrost.
Lezard: Bifrost, the rainbow bridge?
Mystina: Right. The single connection between Asgard and Midgard. I'd like to stand atop the summit of Yggdrasil, where it is said that Odin hung as he receives his wisdom...
Lezard: The only thing that can satisfy your wants and desires is knowledge, hmm? Somehow I'm not surprised, Mysty.
Mystina: What, are you saying you're any different? At least I don't have a...a...Lolita complex!- from Valkyrie Profile: Lenneth
As Lezard notes, Mystina's goal is very fitting for her. It's not that she wants to cross Bifrost and reach Asgard — rather, it's wisdom she seeks. And not just any wisdom, the sum of all knowledge.
Any scholar would want such a thing, but I think in Mystina's case it's about more than the pursuit of knowledge for its own sake. Instead, seeking Bifrost is the only way Mystina knows to further herself. She never got the recognition she felt she deserved at the Flenceburg Academy on account of Lezard, so she set her sights higher.
Though most of Mystina's traits are a front to hide how she really feels, she does seem to consider herself superior to others. In fact, it seems she feels the elves are cheating the rest of the world by hiding Bifrost. She realizes why they protect it, but that's not the only time she reacts that way. She calls Lezard a peeping tom with a Lolita complex when she herself had just been snooping around his tower. So while she considers herself superior, in truth she's not that different from him.
And yet, when Lenneth offers Mystina a place among her einherjar and the chance to study Bifrost, Mystina gives an unexpected response.
Valkyrie: Yes. Also, if you should join us, it may be possible for you to research Bifrost and Yggdrasil.
Mystina: Hmph. Fine. I'll come with you. I mean, it's not as if I really have any other choice, do I?
Valkyrie: Indeed.
Mystina: And you don't have to tempt me with stupid promises, either. Besides, it might be fun to haunt that little freak...- from Valkyrie Profile: Lenneth
I think what Mystina realizes is that her aspirations of reaching Bifrost have no purpose in the company of the gods. Even if she were permitted to go there, the only ones she could prove herself to are the gods themselves. Hers was very much the goal of a mortal, and once she left the material world permanently, she had to give up her material dreams as well.