He claims to have a hair of the Prophet (blessings and peace of Allah be upon him) and he wants to build a mosque in which to preserve it!

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. 

For more information, please see the answer to question no.
147225.

And Allah knows best.