Praying for Jinn Companion to Become Muslim

The hadeeth (narration) you mention is a saheeh (authentic)
hadeeth. It was narrated by Muslim, but there is a difference of scholarly
opinion as to the meaning of the word “fa aslama” in this hadeeth. This
dispute and the most correct view was mentioned by al-Nawawi in his
commentary on this hadeeth. He said: 

With regard to the words of the Prophet (peace and blessings
of Allah be upon him): “There is none of you who does not have a companion
(qareen) appointed for him from among the jinn.” They said, “Even you?” He
said, “Even me, but Allah helped me against him fa aslama [or fa aslamu], so
he only tells me to do that which is good.” 

There are two well-known views that have been reported. Those
who read the phrase as fa aslamu said that it means, “So I am safe [aslamu]
from his evil and temptation.” Those who read it as fa aslama said that it
means, “The qareen became Muslim [aslama] and became a believer, so he only
tells me to do that which is good.” 

They differed as to which view is correct. Al-Khattabi said:
The correct version is fa aslamu [so I am safe]. Al-Qadi ‘Iyad thought that
fa aslama [so he became Muslim] was correct, and this is the preferred
version, because he then said, “so he only tells me to do that which is
good.” And they differed concerning the report that says fa aslama. It was
said that it means he submitted in the sense of surrendering, and it appears
in this form (fa astaslama – so he surrendered) in reports narrated
elsewhere than in Saheeh Muslim. And it was said that it means that he
became a Muslim and a believer. This is the apparent meaning. 

Abu Na’eem al-Asbahani said in Dalail al-Nubuwwah: It was
said aslama meaning he believed. The Prophet (peace and blessings of Allah
be upon him) was the only one whose qareen became a Muslim and a believer. 

Based on this, having one’s qareen become a Muslim was
something that was unique to the Prophet (peace and blessings of Allah be
upon him). 

So it is not prescribed for the Muslim to pray that his
qareen become Muslim, because this is overstepping the limits in du’a
(supplication) by asking Allah for something that belongs only to the
Prophet (peace and blessings of Allah be upon him), and because there is no
report that the sahabah (companions), who were the keenest of all people to
do good and the closest to it, asked Allah to make their qareens become
Muslim, and they did not ask the Messenger (peace and blessings of Allah be
upon him) to pray for that for them when they heard this hadeeth from him.
There is no report that Abu Bakr, ‘Umar, ‘Uthman and ‘Ali (may Allah be
pleased with them), who were very keen to do good, or their sons, did that.
We have to follow the guidance of these great sahabah because they
understood this religion properly and learned it directly from its source,
the Messenger of Allah (peace and blessings of Allah be upon him), and it is
not permissible for us to follow any path but theirs. Allah says
(interpretation of the meaning): 

“And whoever contradicts and opposes the Messenger after the
right path has been shown clearly to him, and follows other than the
believers’ way, We shall keep him in the path he has chosen, and burn him in
Hell — what an evil destination!” [4:115] 

This hadeeth was narrated in the context of warning the
sahabah against the fitnah (trials) of the qareen. Al-Nawawi said: This
hadeeth is a warning against the fitnah of the qareen and his whispers and
temptation. [The Prophet (peace and blessings of Allah be upon him)] warned
us against him so that we can be as cautious as possible. This is what we
must do, according to sharee’ah (Islamic law). 

It is sufficient for us to recite the du’as that Allah has
taught us, for example: 

“And say: ‘My Lord! I seek refuge with You from the
whisperings (suggestions) of the shayateen (devils).

And I seek refuge with You, My Lord! Lest they should come
near me.’” [23:97-98 – interpretation of the meaning] 

And you should recite Surat al-Ikhlas and the Mu’awwadhatayn
(the last two surahs of the Quran) morning and evening, as well as the
dhikrs (words of remembrance) narrated in saheeh ahadeeth (authentic
narrations) from the Prophet (peace and blessings of Allah be upon him). You
should also recite ayat al-Kursiy before going to sleep, and Bismillahi
al-Rahmani al-Raheem before doing anything, and you should seek refuge with
Allah every time you notice the whispers of the shaytan. Allah says
(interpretation of the meaning): 

“And if an evil whisper from shaytan  tries to turn you away
(O Muhammad) (from doing good), then seek refuge in Allah. Verily, He is the
All‑Hearer, the All‑Knower.” [41:36] 

I ask Allah to make you more keen to do good and to protect
us and you from the shaytan and his whisperings. Ameen.