I think the most Biblical part of PHM, outside of the fact that a vessel named after the Virgin Mary carries a man named Grace whose job is to save the world, is the theme of the well-known verse "Greater love hath no man than this, that a man lay down his life for his friends" (John 15:13). Because, yeah, that's what the Hail Mary mission is about. Being willing to lay down your life for the rest of humanity. And it's something that Grace struggles with throughout the story.
I think it's important to note that for most of the story, Grace doesn't have a choice. He's initially snatched up by Stratt and NASA against his will and made to work on a problem he didn't volunteer to work on, even though the work is enjoyable to him. After he's kicked out, he does go back willingly because he cares about his students (there's a hint of the theme there), but even then Stratt puts him in more positions he didn't ask or volunteer for, takes his DNA to test it without asking, drafts him as the backup microbiologist without telling him, and finally sedates and drugs him against his will to get him on the mission. It's the only thing that can save Earth, of course, but it's important to note thematically that this wasn't Grace's choice.
Furthermore, Grace doesn't think much of himself. Even while under the assumption that he did volunteer, he doesn't believe he's truly brave, or that he belongs on the mission, or that he's even that important of a person. He struggles with inferiority and shortcomings. He is a coward. He's lonely. He doesn't have really any friends back home, no significant other, and we don't ever hear anything about his family.
Rocky does it. He is willing to lay down his life for his friend, and very nearly does. It's only after that happens that Grace is able to willingly sacrifice himself, too. Which calls to mind another passage: "There is no fear in love; but perfect love casteth out fear: because fear hath torment. He that feareth is not made perfect in love. We love him, because he first loved us" (1 John 4:18-19).
Grace is able to show his love for Rocky by laying down his life because Rocky did it first. And it's not only because Grace loves Rocky (which is true and important), but because Grace knows he is loved by Rocky. Because of that, he's able to shed his cowardice and self-pity and the mantle of inferiority he's been clinging to and throw himself at the mercy of space for the chance of saving his best friend. He goes willingly, out of love, because of love. And I think there is no more beautiful representation of the kind of love Jesus showed towards us, and the kind of love He wishes us to show towards others.
When you understand that you are loved, it changes your life. It changes your perspective. It changes your motivation. You'll find yourself doing things you never thought you'd do for people you never though you'd do them for. You'll find yourself lost out in the middle of nowhere just to save one soul. And you'll do everything with a smile on your face and a strength in your heart because you are loved.
You'll even rejoice in death.