It seems we can’t know that for sure.
Akhmatova adopted this pseudonym as a teenager since her father didn’t want her to use his last name. Lydia Chukovskaya remembers her conversation with Akhmatova in Записки об Анне Ахматовой (The Akhmatova Journals):
Я спросила, кто придумал ей псевдоним.
— Никто, конечно. Никто мной
тогда не занимался. Я была овца без пастуха. И только семнадцатилетняя
шальная девчонка могла выбрать татарскую фамилию для русской поэтессы.
Это фамилия последних татарских князей из Орды. Мне потому пришло на
ум взять себе псевдоним, что папа, узнав о моих стихах, сказал: «Не
срами мое имя». — «И не надо мне твоего имени» — сказала я.
I asked who came up with her pseudonym.
— No one, of course. No one
took care of me then. I was a sheep without a shepherd. And only a
crazy seventeen-year-old girl could choose a Tatar name for a Russian
poetess. This is the surname of the last Tatar princes from the Horde.
That's why it occurred to me to take a pseudonym for myself, because
Dad, when he found out about my poems, said: "Do not disgrace my
name." "And I don't need your name," I said.
Praskovia Akhmatova was the poet’s great-grandmother. However, as Silver Age scholar Alexandra Chaban comments, she was not a Tatar princess, but a Russian noblewoman (of Tatar origin), and it’s impossible to confirm or disprove any relation to Genghisid rulers (perhaps it was a family legend?). As for the choice of this particular pseudonym, Chaban says it just sounded good:
such a move was not so reckless for the era of the Silver Age: that
time demanded artistic behavior, vivid biographies and sonorous names
from new writers. In this sense, the name Anna Akhmatova perfectly met
all the criteria (poetic — it created a rhythmic pattern in a dactylic
dimeter, and featured an assonance with the "a"; and life-crafting —
it carried a touch of mystery).
One site lists the surnames of her other relatives she could’ve used: Stogova, Motovilova, Evseeva, Voronina. They have much more mundane connotations in Russian: e.g. “Stogova” comes from the word “stack” (as a haystack), and “Motovilova” reminds of the process of winding threads in a loom.
Joseph Brodsky, who knew Akhmatova personally and was sort of under her mentorship, expressed a similar opinion in his introduction to her poetry:
All the same, the five open a’s of Anna Akhmatova had a hypnotic
effect and put this name’s carrier firmly on top of the alphabet of
Russian poetry. In a sense, it was her first successful line;
memorable in its acoustic inevitability, with its Ah sponsored less by
sentiment than by history. This tells you a lot about the intuition
and quality of the ear of this seventeen-year-old girl who soon after
publication began to sign her letters and legal papers as Anna
Akhmatova. In its suggestion of identity derived from the fusion of
sound and time, the choice of the pseudonym turned out to be
ргоphetic.