Praise be to Allah.
It is makrooh to call a child Momin (or Mu’min) for two reasons.
1.It involves self-praise.
2.It may be said in a wrong context. If it is asked, “Is Momin in the house?” or “Have you seen Momin?” [which is Arabic sounds like asking “is there a believer in the house” or “Have you seen a believer?”] and the answer is no, if he is outside the house or has not been seen, this is a disliked usage which may give the impression that there is no believer in the house or that he has not seen any person who is a believer.
There are reports which show that it is not allowed to use names which imply self praise or may be used in a wrong context. Al-Bukhaari (192) and Muslim (2141) narrated from Abu Hurayrah that Zaynab’s name was Barrah (which means righteous) and it was said that she was praising herself, so the Messenger of Allah (blessings and peace of Allah be upon him) named her Zaynab.
And Muslim narrated (2142) that Muhammad ibn ‘Amr ibn ‘Ata’ said: I called my daughter Barrah, but Zaynab bint Abi Salamah told me that the Messenger of Allaah (blessings and peace of Allah be upon him) had forbidden this name. (She said) I was given this name, but the Messenger of Allaah (blessings and peace of Allah be upon him) said: “Do not praise yourselves, for Allaah knows best who among you is righteous.” They said: What should we call her? He said: “Call her Zaynab.”
And Muslim (2136) narrated that Samurah ibn Jundub said: The Messenger of Allaah (blessings and peace of Allah be upon him) forbade us to give our slaves four names: Aflah (prosperous), Rabaah (profit), Yasaar (wealth) and Naafi‘ (beneficial).
This was also narrated by al-Tirmidhi (2836) with the words: “Do not call your slaves Rabaah (profit), Aflah (successful), Yasaar (wealth) or Najeeh (successful).” It was said: Is it a sin? It was said: No.
Al-Nawawi (may Allah have mercy on him) said: What these hadeeths mean is that bad or disliked names should be changed to good ones and there are hadeeths which prove that the Prophet (blessings and peace of Allah be upon him) changed the names of many of the Sahaabah, and he explained the reason in two cases: where it has to do with the meaning, and where it implies self praise or the fear that the name may be used in a wrong context. End quote from Sharh Muslim.
With regard to the name of Allah, which is al-Mu’min with the definite article al-, Allah says (interpretation of the meaning): “He is Allaah, beside Whom Laa ilaaha illa Huwa (none has the right to be worshipped but He), the King, the Holy, the One Free from all defects, the Giver of security [al-Mu’min], the Watcher over His creatures” [al-Hashr 59:23].
It is permissible to call your son ‘Abd al-Mu’min.
‘Abd is not one of the names of the Prophet (blessings and peace of Allah be upon him). Even if we assume it is one of his names, it is incorrect to say Muhammad al-Mu’min or Muhammad Allah or Muhammad al-Raheem if what is meant is ‘Abd al-Mu’min or ‘Abd-Allah or ‘Abd al-Raheem.
To sum up, you should call your son ‘Abd al-Mu’min if you want to have this meaning in his name. Otherwise you should give him some other good name in which there is no self praise or fear that it may be used in a wrong context.
See also the answer to question no. 7180.
And Allah knows best.