Is it essential that the du’aa’s recited in prayer be proven in the Sunnah?

There is nothing wrong with a worshipper saying whatever
du’aa’ he wants in his prayer, those that are narrated in the Sunnah – which
are the best and most comprehensive of du’aa’s – or any kind of permissible
du’aa’ that he wishes. 

The one who recites du’aas that are narrated in sharee’ah
will be rewarded for following what is narrated in sharee’ah, and he will be
rewarded for his du’aa’. 

Among the texts which show that du’aa’ in general is
permissible is the words of the Prophet (peace and blessings of Allaah
be upon him): “Then let him choose whatever supplications he wishes.”
According to another report: “Then let him choose whatever du’aa’ he likes
and say it.” Narrated by al-Bukhaari (835) and Muslim (402). This was
narrated concerning du’aa’ before the salaam at the end of the prayer. 

Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allaah have mercy on him)
said: 

It is prescribed for the believer to say du’aa’ during his
prayer in the places when du’aa’ is allowed, whether the prayer is
obligatory or supererogatory (naafil). The places where du’aa’ is allowed
are prostration, between the two prostrations and at the end of the prayer
after the tashahhud and blessings upon the Prophet (peace and blessings
of Allaah be upon him) and before the tasleem, as it is proven that the
Prophet (peace and blessings of Allaah be upon him) used to pray for
forgiveness between the two prostrations and it is proven that he used to
say between the two prostrations: “Allaahumma ighfir li wa’rhamni
wa’hdini wa’jburni wa’rzuqni wa ‘aafini (O Allaah, forgive me, have
mercy on me, guide me, enrich me, grant me provision and keep me safe and
sound).  

And he (peace and blessings of Allaah be upon him) said:
“As for bowing, glorify the Lord therein, and as for prostrating, strive
hard in du’aa’ therein because it is more likely that you will receive a
response.” Narrated by Muslim in his Saheeh. Muslim also narrated
from Abu Hurayrah (may Allaah be pleased with him) that the Prophet
(peace and blessings of Allaah be upon him) said: “The closest that a person
is to his Lord is when he is prostrating, so say a lot of du’aa’ then.” 

In al-Saheehayn it is narrated from ‘Abd-Allaah ibn
Mas’ood (may Allaah be pleased with him) that when the Prophet (peace
and blessings of Allaah be upon him) taught him the tashahhud he said: “Then
let him choose whatever supplications he wishes.” According to another
report: “Then let him choose whatever du’aa’ he likes and say it.” 

There are many ahaadeeth that speak of this, and they
indicate that it is prescribed to say du’aa’ in these places in the prayer,
whether it has to do with the Hereafter or with worldly matters, so long as
the du’aa’ does not involve any sin or severing of family ties. And it is
better to say a lot of du’aa’s that were narrated from the Prophet
(peace and blessings of Allaah be upon him). End quote. 

Fataawa al-Shaykh Ibn Baaz
(11/171, 172). 

He (may Allaah have mercy on him) also said: 

There is nothing wrong with saying du’aa’ during prayer,
whether that is for oneself or for one’s parents or anyone else. Rather that
is prescribed, because the Prophet (peace and blessings of Allaah be
upon him) said: “The closest that a person is to his Lord is when he is
prostrating, so say a lot of du’aa’ then.” … Then he quoted the ahaadeeth
that we have mentioned above, then he said: 

So if he says du’aa’ when he is prostrating or at the end of
the prayer, for himself or for his parents or for the Muslims, there is
nothing wrong with that, because of the general meaning of these ahaadeeth
and others. End quote. 

Fataawa al-Shaykh Ibn Baaz
(11/173, 174). 

And Allaah knows best.