It may not be possible to completely disentangle all of the factors which lead to the young Lucila Godoy Alcayaga to use a pseudonym. Probably there were several different influences at work.
She is often described as being a very shy woman, and in her early teens when she started publishing, it would be natural for her to try and conceal her identity. The first pseudonyms that she tried - "Soledad" ("loneliness"), "Alma" ("soul"), "Alguien" ("somebody") - indeed seem quite characteristic of teenage angst. These attempts were dropped quite quickly, in favour of her final choice "Gabriela Mistral" which differs in that it could plausibly function as a first name and a surname.
As detailed in another answer, the translator Langston Hughes wrote in 1957 that "She did not sign her poetry with her own name, Lucila Godoy y Alcayaga, because as a young teacher she feared, if it became known that she wrote such emotionally outspoken verses, she might lose her job." This is indeed a plausible explanation. She had already encountered trouble because of her outspokenness. She had left school at 14, and had not been admitted to the Escuela Normal because “a chaplain-professor demanded that they rescind her admission, because she could become a "caudillo" [a ringleader, CDS] of the female students.” As a result she had to begin teaching without having a high-school diploma. Mistral wrote later that “The loss does not hurt me today, but all the teachers and professors who would deny me salt and water during the twenty years I was teaching in Chile - and I've made a note of them - know very well how much it cost me to undertake a teaching career without that piece of paper, the diploma, and that signature."
In the mid 1980s a considerable quantity of documents, correspondence, and personal notebooks, some 120 boxes, were brought from where they had been stored in the Library of Congress in Washington to Chile. Elizabeth Horan, a noted Mistral-scholar, has worked with this freshly available material, and in 2023 published the biography Mistral: A life, the first in a proposed three volume set. I have not been able to get hold of a copy unfortunately, but I did manage to contact her, and her opinion is that Mistral wrote under a pseudonym simply to follow the fashion of the time:
Women writers in Chile at that time quite commonly used pseudonyms... The conventions for women using pseudonyms were standard for that time. The idea is or was that women would do well to protect their reputation, their families, reputation, and not appear, especially ambitious or striving.
It was not just women writers who wrote under pseudonyms, of course. Pablo Neruda, a close contemporary of Mistral, is the pseudonym of Ricardo Eliécer Neftalí Reyes Basoalto.
Finally, a fascinating clue is present in Mistral's final book of poetry Poema de Chile. In the poem entitled “Animales”, the ghost (named in another poem as Gabriela) is asked if she has more than one name. She replies:
Si, el que me dieron
y el que me di de mañosa
y el nuevo me mató el viejo.Yes, the one that they gave me
and the one that I cunningly gave myself
and the new one has killed the old.
This could be read as referring to herself. It is certainly the case that her pseudonym "killed the old" - there are Gabriela Mistral schools, foundations, and prizes, but the old name "Lucila Godoy Alcayaga" is largely forgotten.