Update!! Now that I’ve seen it again, I have a better insight.
First, a correction: he does not put his beanie on while talking to Stratt for the first time, he puts on his helmet. (Putting your guard up, anybody?) He is unsure, wary of her and her intent, and he makes that known, tries to shut her down.
A second correction: when Grace is informing Stratt of his decision not to go, he is not wearing the beanie at first. He pulls the hat on when he’s telling her she “can’t convince [him], [his] mind is made up”. He thought he had a choice, but it was never going to be that way.
Now, onto the new stuff. That beanie isn’t the only beanie, much less hat featured throughout the movie. I touched on the helmet, but Grace also wears ballcaps, a bucket hat, animal hats, a party hat, and a different beanie whilst in space.
We’ll talk about the space beanie first.
It’s to be of note that Grace is not sent up with the beanie we see in the flashbacks — so he makes do with what he has on hand. That is, he cuts a pant leg off pre-vodka-spree and ties it into a makeshift beanie. He’s actually wearing it when he finds the alcohol. He’s the one inhibiting his choices this time. He’s got zero memories, is 11.9 light years from Earth, with two dead crewmates. What does he have left but to wither and drink?
The next time this particular beanie is worn is when Grace makes the calculations to go back for Rocky, and when he loads up the Beetles and sends them on their way. It’s his choice, yes, but is it really a choice when it’s the person you love most? When the question isn’t “will I sacrifice myself for my best friend?” but “how will I find him?” Because, really, he found his someone to be brave for.
(Interestingly enough, Stratt wears a beanie similar to Grace’s Earth one in the future, after the Beetles return. She’s still dedicated to the project.)
The ballcap is next! I don’t think this hat represents the same loss of autonomy that the beanies do. Here’s why: Grace wears it as he watches everybody having fun, singing karaoke, enjoying themselves while he seems to be the only person unable to do so, work spread out on the countertop in front of him. He goes outside to escape the room, to talk to Stratt, and takes the hat off as he asks about her lack of singing, a friendly icebreaker question, an olive branch. It’s during this conversation that he opens up about his discomfort that everybody seems to be so happy, and asks her about her own discomfort and fears (“must be hard having to ask everybody to.. you know” “actually it isn’t”; “you think you’re going to pull it off?” […] “God willing” “you believe in God?” “It beats the alternative”) only to be shot down. He puts on his hat without another word and goes inside. It’s inside, however, that he is subject to Sign of the Times without the hat. She’s saying “I’m sorry, you’re looking for a friend, for someone with whom you want to be vulnerable, but it can’t be with me.”
He wears the space ballcaps twice. He wears two at once in a short clip of him standing in front of the Golden Records — the thing that kicks him into gear. Coming face-to-face with the Golden Records are part of what triggers him to take the hats off and stop avoiding the corpses in the other room.
(Tucked around this clip is Grace wearing a straw bucket hat, to which I don’t really have anything to say about. Grace gives a different bucket hat [I’m fairly certain] to Yáo’s body before sending him off.)
The next time he wears the ballcap (that same Project Hail Mary ballcap), he’s wearing it on his way to the planet Adrian. What’s more is that Rocky is also wearing one. If we’re keeping with the ballcap=emotional barrier theme, there’s far less of a wall between them now. They aren’t using the long distance of a tunnel connecting their ships with the big, thick xenonite wall blocking them from entering each others’ spaces. Up to that point, they’ve talked almost entirely of science or their mission (and mission-related deaths). It’s not impersonal, but there is a distinct separation that eases very quickly afterwards, particularly once they begin the journey and they hang around more. Forced proximity but both of them are lonely and enjoy each others’ company and neither want the barrier. Rocky may have been the first to chip at that wall, mimicking his movements and recreating Grace’s atmosphere in his side of the tunnel, and further by building the ball, but Grace reciprocated pretty quickly in taking his helmet off (bike helmet core much? Taking it off instead of putting it on) and showing Rocky the mental health room.
(Stratt wears a ballcap on the ship, in front of the room of leaders and scientists. She also wears it when she’s smiling at Grace, the moment Dubois and Shapiro die. She is emotionally distant from everybody, however the existence of the ballcap implies the ability and maybe even want to take it off. Three days. She had three days.)
I know I’m recording these out of order, but I would like to discuss the animal hat(s) next. There is a purple bear, a fox mask, and I believe a rainbow bunny(?). He only wears these while he’s alone, at the beginning. And that’s what I think they represent: loneliness. A bit ironic to have animals portray the lack of another being. Grace is wearing the purple bear hat while he dances with the mop (elegantly dubbed Moppy Ringwald), the fox mask while he tries to figure out who he is, and the bunny hat while he’s drinking and crying after sifting through Ilyukhina and Yáo’s things, including their photos of them with their families. I think it’s loss of connection — and he never needs those hats again because, well, he finds that connection.
Lastly (for now at least ;)), the party hat is pretty self-explanatory. Just like how Grace had worn the ballcap and taped the other to Rocky’s ball, and like he made the space beanie himself, he creates the party hats for both Rocky and himself on board. He even goes so far as to draw and colour them. It’s the only hat that doesn’t really have any sort of negative connotation, born of celebration, even if they’re upset at the thought that they’ll never see each other again.
So there we have it! My updated hat analysis. I may be reaching, but this is my goldilocks zone /silly. Don’t be afraid to debate or add on! I’m writing this all at once so I hope this is readable lol