Columbina Lore you Cute Dork. <333!!!!
We currently don't have that much information on Columbina, but there are a lot of speculations. The first section is a recap of what we know, and the second is about theories, general interpretations, and my personal interpretations.
(feel free to skip this section is you just want to read the theories and interpretations of this information)
According to Nahida, the harbingers ranked number three and above have power equal to, or above that of gods. We don't know what her powers may be, but I have some speculations.
Columbina is shown to be entirely uncaring and apathetic during A Winter Night's Lazzo, as most other harbingers at least showed a little bit of respect at the funeral, but Columbina was draped across the coffin, singing like a child. She clearly understood that La Signora was dead, but she simply did not care. She even went as far as to "compliment" Dottore, even though she knew that he wouldn't take it as a compliment.
Her symbol, as shown in the harbinger wheel, is a crying dove.
She has six wings on her head, and she hasn't been shown to open her eyes, even while standing. She was clearly awake, but she chose to not open them.
Theories and interpretations
Speculations about her power:
Given what we have seen about her, I speculate that in game, she will most likely wield a cryo vision. It may be a bit on the nose, as she is a harbinger (centered in Snezhnaya), but her personality seems to fit that of cryo characters. With how apathetic she is, she is most definitely cold on the outside. There are many ideas about her backstory, so we don't know if she experienced a significant loss, but she may have been subjected to truths that caused her to be so uncaring, resulting in the loss of her old self.
I personally think that she is going to be a catalyst user, dealing cryo damage from afar, without exactly exerting herself. She simply doesn't care enough to be more upfront. I think her next option would be a polearm, as it seems to suit her (and if she had a scythe polearm like arlecchino, it'd be a nice contradiction from her Seraphim inspiration). A sword or a bow could suit her, but I'm unsure. She has the name The Damselette, and Wanderer (in his About Damselette voiceline) stated "a 'damsel' who is oblivious and innocent at any given time, and unconcerned and unfeeling in any given situation", so leaning into that "damsel in distress" charade, I believe that a catalyst would be the best choice, to push the idea of being weaker, and being unable to engage in physical combat, even though that is likely not the case.
Her elemental burst may involve her opening her eyes, as it may allow her to do more damage or inflict a status effect upon her opponents.
One common theory about her abilities is that she may be able to freeze people. She might be able to stop them in their tracks (without her cryo abilities, but out of fear), and people assume that it may be related to her abilities shown in Hoyofair, where she has many wings and large eyes, which might be something she uses to "freeze" enemies.
We can deduce from the way she acted during La Signora's funeral and existing voicelines about her that Columbina, on top of being entirely uncaring, is extremely unpredictable, and most likely insane. Even Childe said that he would test his strength with anyone, but not her, because something doesn't feel right. Given that, and her interaction with Dottore, she probably says things to get under people's skin, or to creep them out.
Her constellation gives us an idea of how she wants others to perceive her. She most likely paints herself as a "damsel in distress", and uses that to further her personal goals, before casting away whoever had helped her.
(Credit for this idea goes to the poster of the theory I linked for her constellation) At the beginning of the book Before Sun and Moon, the first section is titled "When the Dove Held Branches", and with Columbina's constellation being a dove, it may have a connection to that. The poster connects Columbina to a Seele, speculating that she may be one of the Seele race, who escaped the fate brought upon others of her kind (similarly to the Goddess of Flowers from Sumeru), and kept her mortal form.
The Seraphim Inspiration:
Columbina is quite clearly based on a Seraphim, the highest ranked angel in Christian, Jewish, and Islamic religion, which are depicted to have six wings, similarly to Columbina. They are stated to circle around God's throne, singing "Holy, Holy, Holy", and proclaiming his greatness to anyone who comes near. With Hoyoverse games, this likely has a deep meaning. The "God" in Teyvat is most likely the Primordial One, or the First Who Came. One speculation is that she was a cherished "angel" (it's unsure as to what name she possesses there, but she was under the Primordial One) who did just that, but gained some knowledge that broke her and caused her to be uncaring and to resent Celestia. Seraphim angels are also stated to be kind, and to help, but the "truth" that Columbina may have uncovered caused a mental break of sorts, resulting in her current personality and occupation.
Columbina in the Commedia Dell'arte:
All of the Fatui Harbingers are named after Characters in the Commedia Dell'arte, with Columbina being no exception. The character she is based on, Columbine (meaning Little Dove), is a stock character (a character with a personality/story that is easily recognisable across many forms of media) that is titled the Tricky Slave, who is an intelligent and cunning character who furthers the relationship between those above them, ensuring that they reach a happy ending.
She was often shown to be the most practical and intelligent character on stage.
The name, Columbine, ended up becoming the most common term for soubretta characters, who are female servants that are often flirtatious in nature.
I don't think that the flirtatiousness will carry over into Columbina (at least in its entirety), but I do think that the fact that she was often depicted as the most intelligent and down to earth character on stage has meaning. It may be a way of showing her knowledge of the speculated "truths" I mentioned earlier, truths beyond typical mortal knowledge/comprehension. She might be the most "enlightened" out of the harbingers, the one who knows the most about the inner workings of Celestia, and the one who knows the most about what is beyond the fake sky.
While we don't have much information on her, I tried to compile what we do know and some speculations into this one post. If there's any wrong information or if there's anything I missed, please let me know and I will try to add more! She's honestly my favorite fatui harbinger (sorry Childe I still love you), and she's so shrouded in mystery that it's just so interesting to research.