Praise be to Allah.

Firstly: 

Tabarruk (seeking barakah or blessings) from the relics of the Prophet (blessings and peace of Allah be upon him) was something that was done at the time of the Prophet (blessings and peace of Allah be upon him), with things such as water left over from his wudoo’, his garment, his food and drink, his hair, and anything else from him. 

As for seeking barakah from anything that had touched his body, such as wudoo’ water, sweat, hair and so on, this is something that was well-known and regarded as permissible by the Sahaabah (may Allah be pleased with them) and those who followed them in truth, because of the goodness and barakah that there is in that, and the Prophet (blessings and peace of Allah be upon him) approved of them doing that. 

It says in al-Mawsoo‘ah al-Fiqhiyyah (10/70): 

The scholars are unanimously agreed that it is permissible to seek barakah from the relics of the Prophet (blessings and peace of Allah be upon him), and the scholars of seerah (Prophet’s biography), shamaa’il (Prophet’s character) and hadith narrated many reports that speak of the noble Sahaabah (may Allah be pleased with them) seeking barakah from various relics of the Prophet (blessings and peace of Allah be upon him). End quote. 

Please see the answer to question no. 10045 

Secondly: 

This tabarruk (seeking barakah) is not permissible with anything other than the relics of the Prophet (blessings and peace of Allah be upon him). So it is not permissible to draw an analogy with anyone else. Seeking barakah from the relics of anyone else among the righteous is a reprehensible innovation (bid‘ah) and it is a means that leads to shirk. 

Ibn ‘Uthaymeen (may Allah have mercy on him) said: 

The Sahaabah used to seek barakah from the sweat of the Prophet (blessings and peace of Allah be upon him), and they used to seek barakah from his saliva, and they used to seek barakah from his garment, and they used to seek barakah from his hair. But in the case of anyone other than him (blessings and peace of Allah be upon him), no barakah is to be sought from any of these things. So one cannot seek barakah from a person’s garment, hair, nails or anything else of his, except in the case of the Prophet (blessings and peace of Allah be upon him). End quote.

Sharh Riyaadh as-Saaliheen (4/243) 

Thirdly: 

There is no proof that any of the relics of the Prophet (blessings and peace of Allah be upon him) exist now. Anyone who claims that any such thing exists has no proof to that effect. Based on that, it is not permissible for anyone to claim to have in his keeping any of the relics of the Prophet (blessings and peace of Allah be upon him), unless he has definitive evidence, and how could he have that? 

The great scholar and historian Ahmad Pasha Taymoor said: 

There is no valid proof for the hairs that were kept by people after that. Something of that which was shared out among the Companions (may Allah be pleased with them) may have reached them, but it is difficult to know what is genuine and what is not. End quote. 

Al-Athaar an-Nabawiyyah by Ahmad Pasha Taymoor (91) 

The great scholar and muhaddith Muhammad Naasir ad-Deen al-Albaani (may Allah have mercy on him) said: 

However we should point out that we believe that it is permissible to seek barakah from the relics of the Prophet (blessings and peace of Allah be upon him) and we do not object that, contrary to what our opponents say about us. 

But this seeking of barakah is conditional upon having the faith that is prescribed and is acceptable to Allah. If a person is not a true Muslim, then Allah will not bring about any good for him by means of this seeking of barakah on his part. Similarly, it is stipulated that the one who wants to seek barakah should have in his possession one of the relics of the Prophet (blessings and peace of Allah be upon him) and use it. 

But we know that his relics (blessings and peace of Allah be upon him), such as his garment, hair and so on, have been lost, and it is not possible for anyone to prove that any such thing exists on a definitive and certain basis. As that is the case, seeking barakah from these relics has become irrelevant in our times and it has become a purely theoretical matter, so we should not speak at length about it.

End quote from at-Tawassul: Anwaa‘uhu wa Ahkaamuhu (144) 

For more information, please see at-Tabarruk: Anwaa‘uhu wa Ahkaamuhu, by Dr Naasir al-Judayyi‘ (156-260) 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: 

What is the ruling on seeking barakah from the relics of the Prophet (blessings and peace of Allah be upon him) after his death, such as his hair and the like? 

He replied: 

The answer to that is that it is not possible to prove that this is a hair of the Messenger (blessings and peace of Allah be upon him) at all. What is said about it being in Egypt in a place where relics were collected is not sound, and it does not exist. 

It is not known that the Sahaabah (may Allah be pleased with them) paid attention to this matter, apart from what is reported from Umm Salamah (may Allah be pleased with her), that she had some of the hairs of the Prophet (blessings and peace of Allah be upon him) in a vessel of silver, and if anyone fell sick he would come to her and she would pour water on it and collect the water, then he would drink it. 

Based on that, it is not possible to prove that this is actually the hair of the Messenger (blessings and peace of Allah be upon him). The most important “relics” are the texts and reports Islam (as opposed to physical relics). As for the physical relics, they are mere relics to which the heart may incline and love them, but what really matters is the teachings of Islam that he left behind. End quote. 

Duroos li’sh-Shaykh al-‘Uthaymeen (2/64) 

Please see also the answers to questions no. 91969, 100105, 204831 

If a person truly wishes to seek barakah in his religious and worldly affairs, then he should follow the Prophet (blessings and peace of Allah be upon him) both outwardly and inwardly, adhere to his guidance, and refrain from that which he prohibited of words, deeds and beliefs. In that is all goodness and blessing (barakah). 

Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: 

When the Prophet (blessings and peace of Allah be upon him) came to the people of Madinah, he brought great blessing to them, as they believed in him and obeyed him, and by virtue of that blessing they attained happiness in this world and the Hereafter; in fact every believer who believes in the Messenger and obeys him will attain something of the blessings (barakah) of the Messenger by virtue of his faith and obedience, and will attain thereby goodness in this world and the Hereafter, to an extent that only Allah knows. End quote. 

Majmoo‘ al-Fataawa (11/113) 

Shaykh al-Albaani (may Allah have mercy on him) said: 

Although the Prophet (blessings and peace of Allah be upon him) approved of the Sahaabah (may Allah be pleased with them) seeking barakah from his relics and touching them during the campaign of al-Hudaybiyah and on other occasions, that was for an important purpose, especially on that occasion; that purpose was to scare the kuffaar of Quraysh and to highlight the extent of the Muslims’ devotion to their Prophet and their love for him, and their willingness to strive hard in serving him and showing him respect. But what we cannot overlook or conceal is the fact that after that campaign, the Prophet (blessings and peace of Allah be upon him) discouraged the Muslims, in a wise and gentle manner, from seeking barakah in this way, and he instructed them to do righteous deeds, which would be better for them before Allah, may He be glorified and exalted, and more beneficial. This is what is indicated by the following hadith: 

It was narrated from ‘Abd ar-Rahmaan ibn Abi Qarraad (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) did wudoo’ one day, and his Companions began wiping themselves with his wudoo’ water. The Prophet (blessings and peace of Allah be upon him) said to them: “What made you do that?’ They said: Love for Allah and His Messenger. The Prophet (blessings and peace of Allah be upon him) said: “Whoever wants to love Allah and His Messenger, or wants Allah and His Messenger to love him, let him be truthful when he speaks, and let him fulfil his trust when he is trusted, and let him be kind to his neighbour.” This is a proven hadith with several chains of narration and corroborating evidence in the Mu‘jams of at-Tabaraani and elsewhere. End quote. 

At-Tawassul (p. 145) 

To sum up: this man’s claim that he has a hair of the Prophet (blessings and peace of Allah be upon him) is a claim for which there is no proof; rather it is a mere claim and it is not permissible for anyone to accept it or believe what he says about it, especially concerning such a matter. If it were true, it would not be a secret; rather it would be widely known, there would be reason to transmit reports concerning it, its isnaads would be examined and verified, and the matter would be something well-known. 

Fourthly: 

His intention to build a mosque and put this hair in it is a corrupt plan and is to be rejected; by doing this he is only following the ways of the people of the Book and diverting people thereby from the straight path of Allah.

 The problem of those people was exaggeration which mades such things justifiable in their minds and caused them to follow the path of innovation and misguidance. Al-Bukhaari (427) and Muslim (528) narrated from ‘Aa’ishah, the Mother of the Believers, that Umm Habeebah and Umm Salamah mentioned a church that they had seen in Abyssinia in which there were images, to the Prophet (blessings and peace of Allah be upon him ). He said: “Those people, if there was a righteous man among them and he died, they would build a place of worship over his grave and put those images in it. They will be the most evil of mankind before Allah on the Day of Resurrection.” 

If these are the most evil of mankind before Allah on the Day of Resurrection, because of that deed, when they were certain about the body of the Prophet whose grave they made into a place of worship, then how about one whom the Shaytaan deceives about a matter of which he does not know its soundness and he does not know any chain of narration on the basis of which he could take it as proof, but he does something for it that is rejected and was not prescribed by the Prophet (blessings and peace of Allah be upon him) for his ummah; rather he forbade them to do such things and warned them against following that path. 

How come the Sahaabah and Taabi‘een did not pay attention to this great  virtue?! Why did they not preserve the relics of the Prophet (blessings and peace of Allah be upon him) and hand them down from one to another so as to preserve them, and put some of them in the mosques? 

Conclusion: 

Seeking barakah from this hair is not permissible and it is a means that may lead to shirk, because it is impossible to prove that it is actually a hair of the Prophet (blessings and peace of Allah be upon him). By the same token, it is not permissible to build a mosque for it and keep it in that mosque, even if we assume for argument’s sake that it is actually a hair of the Prophet (blessings and peace of Allah be upon him) and is proven by an unbroken, sound chain of narration.