The manifestation of Allah's Name ' Al-Mu'min' in the month of Ramadan: 
 
Allah's Name ' Al-Mu'min' is mentioned in the following Noble Ayah in which He says:

"He is Allah than Whom there is La ilaha illa Huwa (none has the right to be worshipped but He) the King, the Holy, the One Free from all defects, Al-Mu'min ( the Giver of security), the Watcher over His creatures, the All-Mighty, the Compeller, the Supreme. Glory be to Allah! (High is He) above all that they associate as partners with Him."

[Al-Hashr, 23]

The meanings of Al-Mu'min:

Actually, believers (Al-Mu'minun) are called so because they believe in Allah the Almighty, (His angels, His books), His Messengers, and the Last Day. However, what does Allah's Name 'Al-Mu'min' mean?
Linguistically, "mu'min" is a word that describes the doer of the act: 'amina' or 'āmana' which has two nominal verbs (masdar): Amn and Ama'n.
Let's ponder over the meanings of the verb: 'āmana'
The first meaning of the verb 'āmana':
The first meaning of the verb 'āmana' is to believe, Allah the Almighty says:

"They said: "O our father! We went racing with one another, and left Yusuf (Joseph) by our belongings and a wolf devoured him; but you will never believe us [the transliteration of this phrase is: bi mu'minin lana] even when we speak the truth."

[Yusuf, 17]

The second meaning of the verb 'āmana':
 
The second meaning of the verb 'āmana' is to make one feel safe and secure. Allah the Almighty says:

"(He) Who has fed them against hunger, and has made them safe [the transliteration of this phrase is: wa āmanahum] from fear."

[Quraysh, 4]

As for Allah's Name 'Al-Mu'min', it means He supported His Messengers with miracles to prove their truthfulness.
In other words, Allah the Almighty sent down many Noble Messengers and Prophets, such as our Prophet, may Allah's Peace and Blessings be upon him, Prophet Musa PBUH, Prophet Isa PBUH, etc. and He supported them with signs and miracles to make people believe in them.
Also, Al-Mu'min is the One Who fulfills His Promises to the believers; Whose Actions prove the truth of His Word. For example, Allah the Almighty has promised true believers great bounties in both this life and the Hereafter, so when you abide by His Orders and Prohibitions, He will fulfill His Promise for you and you will live a good life.
  Furthermore, all the aspects of your everyday life confirm that Allah's Promises, which are mentioned in the Noble Qur'an, are true. For example, when you adhere to the Islamic teachings and rulings on buying and selling, you will find peace of mind, Allah the Almighty will grant you lawful and good sustenance, and your business will succeed. Also, when you are honest and faithful, you will gain a good reputation. Thus, when you stick to Allah's Commands, Allah the Almighty will fulfil His Promises to you, and all the aspects of your life will confirm that Allah's Promise is true.
Also, Al-Mu'min is the One Who grants mankind security (by making the materials' chemical and physical properties unchangeable and immutable).
For example, if the properties of iron are changeable; if sometimes it were hard but at other times it were soft, all buildings would have collapsed and people would never have been able to construct any building. So, the fixity of iron's properties makes people feel safe and secure; people live in high building and they do not fear that the buildings they live in may collapse, since the properties of iron will never change. 
 Thus, the immutability of materials' properties grant people a feeling of safety and security.
 

The true believer's connection with 'Al-Mu'min'

As for the application of Allah's Name "Al-Mu'min" in the believer's life, his actions should correspond with his words and should match them. Besides, people should feel safe from any harm (i.e. hurtful acts or words) caused by him. 
 Abu Hurairah, may Allah be pleased with him, narrated that the Prophet, may Allah's Peace and Blessings be upon him, said:

 "The Muslim is the one from whose tongue and hand people are safe, and the believer is the one from whom people's lives and wealth are safe."

[At-Tirmidhi and An-Nasa'i]

In conclusion, the sacred month of Ramadan provides a huge opportunity for the true believer to increase his Iman (faith) and to deepen his connection with Him. Moreover, Allah's Name "Al-Mu'min" is manifested in the way He makes His worshipper feel safe and secure when they perform acts of obedience to Him. It is also manifested in the way He reassures them that He will keep them safe from His Punishment and torment.