The Emancipator, The Fulfiller of Needs, The Bestower of Wealth.
Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is Al-Mughni (in Arabic: ٱلْمُغْنِيُّ), He is The One who gives abundantly and provides all that is needed. He enriches all of creation so that they may be satisfied and content.
From the root ghayn-nun-ya (غ ن ي) which has the following classical Arabic connotations: to be free from wants or needs, to be self-sufficient, independent, to be able to do without help from others, to be content, satisfied to be rich, wealthy, flourishing.
Two names are often mentioned in pairs, al-Ghani (The Rich/The Self Sufficient) and al-Mughni (The Enricher). You should read the writeup for the name al-Ghani to more deeply understand this name al-Mughni. As a note, al-Ghani is a name mentioned in the Qur'an 18 times, whereas al-Mughni is not widely recognized by scholars as it's not mentioned in the Qur'an.
To provide a brief summary, the name al-Ghani implies that Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is free from wants, independent, and self-sufficient. It's because of this nature of being fully satisfied and abundant that makes Him al-Mughni, the enricher. He is not only rich in a monetary sense but in all regards. He is the one needed and whose riches all others depend on. Al-Mughni is the ultimate fulfiller of needs.
There are two aspects of this, ask from Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ when you are needy. But to also not deny or belittle the blessings He's already given (see al-Ghani for further clarification). There is a belief among scholars that there is a methodology or certain things a person can do to increase the chance of their du'a being fulfilled. Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ says in Surah Baqarah, "[...] I respond to the invocation of the supplicant when he calls upon Me. So let them respond to Me [by obedience] and believe in Me that they may be [rightly] guided." (Qur'an 2:186) Allah is near to those who call upon Him, but He asks in return that we be obedient to what He has asked of us. This is part of the concept of tawassul. By increasing your nearness to Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ with actions beloved to Him, you increase the likelihood of obtaining His favor. After Prophet Zakariya (as) made du'a, we're told in the Qur'an, "So We responded to him, and We gave to him John, and amended for him his wife. Indeed, they used to hasten to good deeds and supplicate Us in hope and fear, and they were to Us humbly submissive." (Qur'an 21:90)
Another shift we can make is going from an egocentric view to making a du'a in complete submission to Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ. The idea is sometimes we make our du'a, and we only focus on the self. I need this, I need that. Please make this happen. There is nothing wrong with this, Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is al-Mughni, and there is nothing He can't do. But note in the du'a of Prophet Zakariya (as), "My Lord, grant me from Yourself a good offspring. Indeed, You are the Hearer of supplication." (Qur'an 3:38) and also, "My Lord, do not leave me alone [with no heir], while you are the best of inheritors." (Qur'an 21:89) The importance is placed on seeking it from the one who can provide you it. Both du'as begin with exalting Him, Rabbanna "My Lord," all things come from yourself so grant me this, you're the hearer of supplication, you're the best of inheritors so provide for me.
Ammany-yujeebul mud tarra izaa da’aahu wa yakshifus sooo’a wa yaj’alukum khula faaa’al ardi ‘a-ilaahun ma’allah qaleelan maa tazak karoon Ammany-yahdeekum fee zulumaatil barri wal bahri wa many yursilu riyaaha bushran baina yadai rahmatih; ‘a-ilaahun ma’allah; Ta’aalal laahu ‘ammaa yushrikoon
"Is He [not best] who responds to the desperate one when he calls upon Him and removes evil and makes you inheritors of the earth? Is there a deity with Allah? Little do you remember. Is He [not best] who guides you through the darknesses of the land and sea and who sends the winds as good tidings before His mercy? Is there a deity with Allah? High is Allah above whatever they associate with Him." — (Qur'an 27:62-63)
We can also use our state or circumstance and say, "Oh Allah, I am faqeer (فقیر)," meaning from poverty or in a dire state. From this ayah, we learn that Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ loves to respond to the desperate cry of His believer. The subtle spiritual shift is from revolving around your needs to sincere submission and reliance upon Allah. Instead of making the du'a about yourself, make it about who you're asking from. Directing your heart in neediness to receive from Him and His beautiful attributes. Of course, following these criteria is not always necessary. Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is most merciful and gives to whom he wills. But we should be aware of the actions beloved to Him and seek His approval through such manners.
Wa annahoo huwa aghnaa wa aqnaa
"And that it is He who enriches and suffices." — (Qur'an 53:48)
Another example of proper du'a etiquette is shown in the story of Musa (as). After escaping Egypt to avoid persecution, he traveled through the desert until he came across a group of male shepherds watering their flocks at a well. Keeping their distance, he noticed two women waiting in the back with their sheep. Why? For whatever reason, they felt safer waiting than mixing with the men who were gathered at the well. They explained to Musa (as) that their father was an older man and unable to feed their flock, so they had to take on this duty. Now keep in mind that Musa (as) himself was disheveled and exhausted from his travels. He likely did not have much to eat or to drink. Still, he empathized with the girls, "So he watered (their flocks) for them; then he turned back to the shade…." As he turned back to the shade, he recited the following du'a.
Fasaqaa lahumaa summa tawallaaa ilaz zilli faqaala Rabbi innee limaaa anzalta ilaiya min khairin faqeer
"So he watered [their flocks] for them; then he went back to the shade and said, 'My Lord, indeed I am, for whatever good You would send down to me, in need.'" — (Qur'an 28:24)
He was in a dire state and made this plea before Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ to send down more good. He knew he was impoverished and saw that the opportunity which presented itself was a gift from Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ and he was in need of more. It's a beautiful du'a that asks from Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ but also keeps in mind the role we play as well.
The ways in which the believer can try to connect with this name is to spend His time busying himself in the service to others. You can enrich your soul by trying to enrich the lives of others. We can see this quality through stories of Prophet Muhammad ﷺ.
Jabir b. Abdullah reported: It never happened that Allah's Messenger ﷺ was asked for anything and he said: No.  The Prophet Muhammad ﷺ character was perfect. Whenever someone came to him asking for help, he could not find in his heart to turn them away. He would spend his time being of service, which is part of what made him so loved among the companions. He was there for them, and they knew it.
Anas b. Malik (ra) reported: I served the Messenger of Allah ﷺ for ten years, and, by Allah, he never said to me any harsh word, and he never said to me about a thing as to why I had done that and as to why I had not done that.  The Prophet ﷺ never tried to spread negativity, hatred, or curse those around him. He never questioned why Anas ibn Malik (ra) did something or did not do something. This does not mean he was passive, either. All we can infer from this is the supreme leadership skills he possessed. What reputation follows us? How would someone speak of working with you after spending 10 years together? Did you try to uplift others? Were you supportive? Or would they say you complained frequently and brought everyone down?
We can never be truly devoid of needs; we are human beings with desires and needs. But as mentioned in al-Ghani, we can work on being deliberate in our worldly desires. If it benefits you, then ask of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ who is al-Mughni. But if you only want something out of greed, pride, or to rise in social rankings, we should think twice about its importance. It's possible to have 1 million dollars and still believe you don't have enough. You may have friends who have boats and multi-million dollar mansions, making you feel less than. This is why it's been famously said it is not greed that drives the world but envy. Today, we have fridges, cars, video games, computers, phones, and on-demand food service. The average person is living better than kings a thousand years ago. But do you feel like a king? In al-Ghani, we mentioned how Umar (ra) cried when he saw how the Prophet ﷺ was living. This house had one bed, a few cloth and pots, black curtains, water skins that hung above his bed, and an uncomfortable mat that left marks on his body. When Prophet Muhammad ﷺ was asked about this, "Won't you be satisfied that they enjoy this world and we the Hereafter?" So it is possible we have been enriched by al-Mughni and not even realized it.
Ali (ra) narrated: that a Mukatib came to him and said: “Indeed I am not capable of my Kitabah so aid me.” He said: “Should I not teach you words that the Messenger of Allah (ﷺ) taught me? If you had a debt upon you similar to the mountain of Sir, Allah would fulfill it for you. He said: 
Allahummakfini bihalalika `an haramika, wa aghnini bi fadlika `amman siwaka
"O Allah, suffice me with Your lawful against Your prohibited, and make me independent of all those besides You"