Prayer from the heart
A person must turn to G-d with all his worries and needs, not only for himself but also for another. Each person can pray to the One G-d with his own words in a language he understands .A “simple” prayer, straight from the depths of the heart ascends to heaven the fastest. We can learn the beauty of a “simple” prayer for the sick from the following history of Rebbe Nachman.
Once, when Rebbe Nachman was ill, he asked his grandson to pray for him. His grandson said, “give me your watch”, as if he was a rebbe taking a pidyon nefesh. The Rebbe said, “you see? He is already a guter yid (a miracle worker)”. He gave his grandson his watch. His grandson said, “G-d! Make Zaydie better”. Everyone else laughed. The Rebbe said, “why are you laughing? That is how we must pray”.
Tehillim – Psalm 20
The Sages did not establish liturgy or prayer times or supplications to G-d for non-Jews. Psalms can always be used for prayer, beause they are all prayers to G-d composed with holy inspiration.
For the sick person you can use for example Tehillim – Psalm 20. You can name the person in the Psalm in the place of “you.” You can combine multiple names of sick people in 1 prayer. It is appropriate to give a proper charity donation and to do other good deeds to benefit the healing of the sick person.
For the leader. A psalm of David.
May the L-RD answer you (name of the person) in time of trouble,
the name of Jacob’s G-d keep you safe.
May He send you help from the sanctuary,
and sustain you from Zion.
May He receive the tokens of all your meal offerings,
and approve your burnt offerings. Selah.
May He grant you your desire,
and fulfill your every plan.
May we shout for joy in your victory,
arrayed by standards in the name of our G-d.
May the L-RD fulfill your every wish.
Now I know that the L-RD will give victory to His anointed,
will answer him from His heavenly sanctuary
with the mighty victories of His right arm.
They [call] on chariots, they [call] on horses,
but we call on the name of the L-RD our G-d.
They collapse and lie fallen,
but we rally and gather strength.
O L-RD, grant victory!
May the King answer us when we call.
As mentioned above, it is not necessary to use a set prayer, but some people like to use a set structure. Below is an example of a set prayer.
When we pray for a sick non-Jew, we look to the paternal side. We can use the following prayer.
For a non-Jewish male: May the Holy One, blessed be He, be filled with mercy for (mention the sick person’s given names), son of (or ben) (mention the given names of the sick person’s non-Jewish father if you know that), to restore him to health and to cure him, to strengthen him and to invigorate him. And may G-d hasten to send him from Heaven a complete recovery to all his bodily parts and veins, a healing of spirit and a healing of body. Amen
For a non-Jewish female: May the Holy One, blessed be He, be filled with mercy for (mention the sick person’s given names), daughter of (or bat) (mention the given names of the sick person’s non-Jewish father if you know that), to restore her to health and to cure her, to strengthen her and to invigorate here. And may G-d hasten to send her from Heaven a complete recovery to all her bodily parts and veins, a healing of spirit and a healing of body. Amen 
Now we don’t always know the father’s name. Then instead of the father’s name you can use the common name of the ancestor/ forfather of all non-Jews: Noah.
When we pray for a Jewish person who is sick, we look to the maternal side. We can use the following prayer.
For a Jewish male: May the Holy One, blessed be He, be filled with mercy for (mention the sick person’s given names), son of (or ben) (mention the given names of the sick person’s Jewish mother if you know that), to restore him to health and to cure him, to strengthen him and to invigorate him. And may G-d hasten to send him from Heaven a complete recovery to all his bodily parts and veins, a healing of spirit and a healing of body. Amen
For a Jewish female: May the Holy One, blessed be He, be filled with mercy for (mention the sick person’s given names), daughter of (or bat)(mention the given names of the sick person’s Jewish mother if you know that), to restore her to health and to cure her, to strengthen her and to invigorate here. And may G-d hasten to send her from Heaven a complete recovery to all her bodily parts and veins, a healing of spirit and a healing of body. Amen 
Now we don’t always know the mother’s name. Then instead of the mother’s name you can use the common name of the ancestor/ formother of all Jews: Sarah.
When praying for a convert, there are two Rabbinical opinions:
1: Since the maternal side of the descent of this convert’s Jewish soul goes directly back to the Jewish foremother Sarah, the person’s name as “So-and-so ben/bat Sarah” is correct in actuality (in spiritual terms), and not only generically.
2: The very first Jew was Abraham, before Sarah was born. So on the soul level, the progenitor of all Jewish souls (including that of Sarah) was Abraham’s soul. Therefore, this opinion holds that it is more precise to refer to a converted Jew as ben/bat Abraham, in any case. (This is de opinion of the minority.) 
Why do these prayers differ?
One possible reason why prayers differ between Jews and non-Jews is the lineage. For non-Jews lineage goes through the father and for Jews it goes through the mother.
A Biblical source for a child being Jewish through the mother is found in Deuteronomy 7:3–4 
You shall not intermarry with them: do not give your daughters to their sons or take their daughters for your sons.For they will turn your children away from Me to worship other gods, and יהוה’s anger will blaze forth against you, promptly wiping you out.
Rabbi Yoḥanan says in the name of Rabbi Shimon ben Yoḥai: As the verse states with regard to the same issue: “Your daughter you shall not give to his son…for he will turn away your son from following Me” (Deuteronomy 7:3–4). Since the verse is concerned that after one’s daughter marries a gentile, the father will lead his children away from the service of G-d, this indicates that your son, i.e., your grandson, from a Jewish woman is called “your son” by the Torah, but your son from a gentile woman is not called your son, but her son.
That it was a custom for Jews to mention their mother during prayer we see in Psalm 116:16
I am Your servant,
Your servant, the son of Your maidservant;
You have undone the cords that bound me 
wherein king David specifies his mother (although his father was also a very righteous person).
That lineage for non-Jews goes through the father we learn from Nachmanides on Leviticus 24:10. were he quotes French scholars (without further identification) who teach that non-Jews descent goes through the father.
We can learn the same from Talmud Yevamot 78b which shows that the nationality of a non-Jew goes through the father (to determine whether he is a Canaanite, Edomite, Egyptian, Amonite, etc.). The French scholars quoted by Nachmanides teach that paternal descent also applied to the Jewish people in earlier times and that this changed when they accepted the covenant at Sinai. Nachmanides himself disagrees with this view, teaching that maternal lineage applies from the forefathers.
We can also see this maternal lineage in Ezra 10:2-3:
“We have trespassed against our G-d by bringing into our homes foreign women from the peoples of the land; but there is still hope for Israel despite this. Now then, let us make a covenant with our G-d to expel all these women and those who have been born to them, in accordance with the bidding of the L-rd and of all who are concerned over the commandment of our G-d, and let the Teaching be obeyed.
Rashi teaches on this that the non-Jews women had to be sent away and their children, which automatically means that the children were also not Jewish.
So, in summary, we can say that from Adam the lineage went through the father and this changed to the lineage throught the mother in the time of the Jewish forefathers Abraham, Izaak and Jakob or at the covenant at Sinai and that may be one of the reasons why the set prayer is different.
But whether you want to use a set prayer or pray with your own words in your own language, do it from the depth of your heart.
May it be that we do not need to use these prayers.
By Angelique Sijbolts
Sources:  Pray as a child  Sefaria  Prayers for Noahides: Community Services & Personal Worship, Why Use the Mother’s Name When Praying for Someone, Prayer for people with physical ailments  Information form Dr. Michael Schulman  Sefaria  Kidushin 68b-3, see also Yevamot 17a:2,  Sefaria  Nachmanides,  Did Judaism Use to Practice Patrilineal Descent?  Sefaria
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