Even though the question is about why Lucila Godoy Alcayaga adopted a pseudonym at all rather than why she chose that specific pseudonym, those questions seem to be intertwined. The sources I consulted don't cite any statements by the poet herself, so theories about what motivated the adoption of the pseudonym are based on assumptions about its meaning.
According to Peter Mayo and Paolo Vittoria (Critical Education in International Perspective, Bloomsbury, 2021), the pseudonym
derives from the names of two poets she must have greatly admired — Gabriele D'Annunzio and Frédéric Mistral.
Note the "must have", which suggest a degree of uncertainty. This seems to be the most widely adopted theory. It is also mentioned in Contemporary Latin American Cultural Studies, edited by Richard A. Young and Stephen Hart (Taylor and Francis, 2014) and The Women of the Nobel by Massimo di Terlizzi (Sem Edizioni, 2014).
According to Paul Burns and Salvador Ortiz-Carboneres, editors of Mistral's Selected Poems (Oxbow Books, 2006, page 10), Lucila Godoy Alcayaga
began using the pseudonym Gabriela Mistral, and this name (made up either of the Archangel Gabriel or the Christian name of Gabriele D'Annunzio plus the surname of Frédéric Mistral) became generally accepted as hers, not just as a nom de plume (and so is now used here).
This introduces an alternative theory for the origin of "Gabriela", which can also be found in other sources. For example, Stephen Smith (An Inkwell of Pan Names, Xlibris US, 2006, page 21) writes,
Some biographers believe that Alcayaga derived her pen name from these two authors [Gabriele D'Annunzio and Frédéric Mistral] whose writing she liked. Alcayaga combined the feminine form of Gabriele with the surname Mistral to form the pseudonym, Gabriela Mistral. Other biographers believe that the author formed her pen name from the name of the divine messenger, the Archangel Gabriel, and from the mistral, a cold, dry, northerly wind whose currents move swiftly over southern France.
This introduces a second potential origin for the name "Mistral", but still doesn't explain why the poetess chose a pseudonym. Smith continues,
What is known is that, as a teenager, Lucila Alcayaga, using a variety of pen names, began contributing poetry to Chilean newspapers. She first used Gabriela Mistral as a pseudonym for a poem published in Elegancias, a French magazine.
Martin C. Taylor mentions her earlier pseudonyms (Gabriela Mistral's Struggle with God and Man. McFarland, 2012, page 24–25):
In 1906, she wrote ten selections, in 1907, eight, and in 1908, thirteen, sanctifying herself as "Alma" ("Soul"), and posturing in a materialistic, secular way as "Alguien" ("Somebody"), the pseudonym that marks the book's opening Epigraph/Manifesto. The assertion of "posturing gains greater validity with regard to "Alguien". Pedro Pablo Zegers B. claims this pseudonym is an anagram—a scrambling of letters—to playfully and furtively involved the initials of her full name, i.e. LGA.
Alcayaga first used the pseudonym Gabriela Mistral in 1908 and according to Taylor, there are three categories of opinions about its origin:
One group held that Lucila combined the given name of the Italian novelist of The Triumph of Death, Gabriele D'Annunzio, with the surname of Frédéric Mistral (1830–1914), the Provençal poet of Miréio (1859), and Nobel laureate (1904). Another group held that she combined Dante Gabriel Rosetti, the English Pre-Raphaelite, and Frédéric Mistral to give rise to the name. Lucila finally cleared up the provenance, in 1946, after years of torturous elaborations and hesitations. She declared, in accord with a third group, that the name derived from a "nombre de arcángel con apellido de viento" ("the first [name] from an archangel, the last, a wind"). (…) She re-fashioned herself in sacred and secular symbols, "Gabriela Mistral" fused materialized divine essence (the archangel), with earthly, but invisible, matter (the mistral).
However, even this last quote explains only the origin; the motivation is Taylor's interpretation.
Finally, there is the following motivation for "mistral" by the poetess, quoted on Poetry Foundation:
I have great love of the wind. I take it for one of the most spiritual of the elements—more spiritual than water. I wanted, then, to adopt a name of wind, but not "hurricane" or "breeze"; one day, teaching geography in my school, I was impressed by the description of the wind made by Reclus in his famous work, and I found in it that name: Mistral. I immediately adopted it as my pseudonym, and this is the true explanation of why I use the last name of the singer of Provence.
In sum, biographers don't seem know for certain why Gabriela Mistral decided to use a pseudonym.