Praise be to Allah.
Making the last prostration of the prayer lengthy for the purpose of offering supplication is not Sunnah; rather it is contrary to the Sunnah.
Al-Bukhari (792) and Muslim (471) narrated that al-Bara’ ibn ‘Azib (may Allah be pleased with him) said: The prayer of the Messenger of Allah (blessings and peace of Allah be upon him), his bowing, when he lifted his head from bowing, his prostration and his (sitting) between the two prostrations were almost all equal in length.
Shaykh Muhammad ibn Salih al-‘Uthaymin (may Allah have mercy on him) said: With regard to making the last prostration longer than the other essential parts of the prayer, in order to offer supplication and pray for forgiveness in it, is there any problem with the prayer if the last prostration is made lengthy?
Making the last prostration of the prayer lengthy is not Sunnah, because the Sunnah is to make all the actions of the prayer similar in length: bowing, rising from bowing, prostrating, and sitting between the two prostrations, as al-Bara’ ibn ‘Azib (may Allah be pleased with him) said: I observed the prayer of the Prophet (blessings and peace of Allah be upon him) and I noticed that his standing, his bowing, his prostration, and his sitting between the tasleem and departure were almost all equal in length. This is what is best. But there is a place for supplication other than prostration, namely the tashahhud. When the Prophet (blessings and peace of Allah be upon him) taught ‘Abdullah ibn Mas‘ud the tashahhud, he said: “Then let him [the worshipper] choose whatever supplication he likes.” So he can make the supplication short or long, after reciting the final tashahhud and before saying the taslim. End quote.
Fatawa Nur ‘ala ad-Darb (tape no. 376, side B).
Shaykh ‘Abdullah al-Jibrin (may Allah preserve him) said:
I do not see any evidence, and I have not read anything to suggest that the last prostration of the prayer should be made lengthy. But perhaps some imams do that to alert people to the fact that this is the end of the prayer, or the last action of the prayer, so they make it lengthy in order for people to realise that the sitting for the last tashahhud is approaching. If that is the case, it is not sufficient justification for making it lengthy. End quote.
Fatawa ash-Shaykh Ibn Jibrin: Ahkam Qat‘ as-Salah, Fatwa no. 2046), from his website.
In Fatawa Islamiyyah (1/258), he (may Allah preserve him) said:
I do not recall any evidence to suggest that the last prostration should be made lengthy. Rather the hadiths speak of making all the essential parts of the prayer equal in length or almost equal in length. End quote.
And Allah knows best.