She has three daughters – is it permissible for her to ask Allaah for a righteous son?

Yes, it is
permissible for you to ask Allaah to bless you with a righteous son. The
Prophet of Allaah Sulaymaan (peace be upon him) wished for that. It was
narrated from Abu Hurayrah (may Allaah be pleased with him) that the Prophet
(peace and blessings of Allaah be upon him) said: “Sulaymaan ibn Dawood
said: Tonight I shall go around to ninety wives and each woman will become
pregnant with a knight who will strive for the sake of Allaah.” Narrated by
al-Bukhaari (3424) and Muslim (1654). But he wished for these boys so that
they would strive for the sake of Allaah and support the religion of Allaah,
not just so that he could enjoy having sons. 

And Zakariya
(peace be upon him) prayed to his Lord and said:  

“O My
Lord! Leave me not single (childless), though You are the Best of the
inheritors”

[Maryam
19:89] 

And Allaah
says that among the prayers of the slaves of the Most Merciful are the words
(interpretation of the meaning): 

“And
those who say: Our Lord! Bestow on us from our wives and our offspring the
comfort of our eyes, and make us leaders of the Muttaqoon (the pious)”

[al-Furqaan 25:74] 

We say “a
righteous son” because a son may be a blessing for his family or he may be a
curse, so we ask Allaah to bless you with a righteous son who will bring you
joy. 

It should be
noted that Allaah gives both male and female children, to whomsoever He
will, by His wisdom and knowledge. The believer can only accept and submit
to the will of Allaah, and rejoice in His blessing. Allaah, may He be
glorified and exalted, says (interpretation of the meaning): 

“To
Allaah belongs the kingdom of the heavens and the earth. He creates what He
wills. He bestows female (offspring) upon whom He wills, and bestows male
(offspring) upon whom He wills.

50. Or He
bestows both males and females, and He renders barren whom He wills”

[al-Shoora 42:49-50] 

Having
daughters is a blessing of which many people are unaware, because they are
easy to raise, and because of the great reward that a person may gain
through them, as the Prophet (peace and blessings of Allaah be upon
him) said: “Whoever takes care of two girls until they reach puberty, he and
I will come like this on the Day of Resurrection” – and he held his two
fingers together. Narrated by Muslim (2631). According to al-Tirmidhi:
“Whoever takes care of two girls, he and I will enter Paradise like this”
and he pointed with his two fingers. Classed as saheeh by al-Albaani in
Saheeh al-Tirmidhi. 

May Allaah
help us all to do that which He loves and which pleases Him. 

And Allaah
knows best.