(The Responsive One)
The Responsive, The Answerer of Prayers, The Hearkener.
Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is Al-Mujeeb/Al-Mujib (in Arabic: ٱلْمُجِيبُ) meaning He is the one very close to His believers and readily available to hear and answer the supplications of those who ask. He is the responder to all needs, invitations, and prayers. Whoever needs help in difficulties, they should call upon this name of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ.
From the root jim-waw-ba (ج و ب), which has the following classical Arabic connotations: to answer, reply, respond, accept to cut, pierce, penetrate, to have a dialog or conference, to comply with a desire when invited to do so.
Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is close to you:
Al-Mujeeb (The Responder) is mentioned in the Qur'an once and is beautifully paired with Al-Qareeb (The Close One). It's because Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is near to us, He responds. Nearness does not have to mean in a literal physical sense. For example, if I asked who is closest to you, would you look to your immediate right? No, you would likely respond with your spouse, a parent, a sibling, or a childhood friend. This person is close to you because they know you on a level most don't. They can be trusted and genuinely care for you and accept you. Similarly, Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is Qareeb (near), and He is Al-Mujeeb (The Responsive).
Wa ilaa Samooda akhaahum Saalihaa; qaala yaa qawmi budul laaha maa lakum min ilaahim ghairuhoo Huwa ansha akum minal ardi wasta’ marakum feehaa fastaghfiroohu summa toobooo ilaih; inna Rabbee Qareebum Mujeeb
"And to Thamud [We sent] their brother Salih. He said, “O my people, worship Allah; you have no deity other than Him. He has produced you from the earth and settled you in it, so ask forgiveness of Him and then repent to Him. Indeed, my Lord is near and responsive." — (Qur'an 11:61)
Although these next few verses don't mention the name Al-Mujeeb directly, they further emphasize this quality of being responsive.
Wa qaala Rabbukumud ooneee astajib lakum; innal lazeena yastakbiroona an ibaadatee sa yadkhuloona jahannama daakhireen (Section 6)
"And your Lord says, “Call upon Me; I will respond to you.” Indeed, those who disdain My worship will enter Hell [rendered] contemptible." — (Qur'an 40:60)
Wa izaa sa alaka ‘ibaadee ‘annnee fa innee qareebun ujeebu da’wataddaa’i izaa da’aani falyastajeeboo lee wal yu’minoo bee la’allahum yarshudoon
"And when My servants ask you, [O Muhammad], concerning Me – indeed I am near. 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)
And from the hadith, it was narrated by Anas: "The Prophet ﷺ said, "My Lord says, 'If My slave comes nearer to me for a span, I go nearer to him for a cubit; and if he comes nearer to Me for a cubit, I go nearer to him for the span of outstretched arms; and if he comes to Me walking, I go to him running.'" 
Addressing the concern of du'a not being responded to:
The Prophets in Islam (may peace be upon them) were regular people like you and me, except we held them to a higher standard as they were impeccable in character. Despite their respected status in Islam, they suffered and endured a lot. Contrast this with when we make du'a, and it's not immediately fulfilled. We may grow impatient or falsely accept that we're not on good terms with our creator or that we're being punished. Yet in the Qur'an, we are told Prophet Nuh (as) made du'as asking for the protection of his wife and son, yet when the tide came, it drowned his family because they were of the deniers (see 11:42-47).
Prophet Yusuf (as) was an innocent child. Do you think he deserved to be thrown into a well by his own brothers and left to die? Prophet Yaqub (as), the father of Yusuf (as), was deeply hurt by the loss of his child. Don't you think he prayed to Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ every day, asking for the safe return of his child? Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ tells us "give good tidings to the patient" (Qur'an 2:155). We know Yaqub (as) du'a was answered as he was reunited with his son. However, this was only after decades of so-called "unanswered prayers."
In learning the attributes of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ we come to understand whatever Allah does, He does good. We may see the challenges we're currently facing as difficult. Still, there is a reward for going through the pain and struggle and remembering Allah through it all. The bigger the hill, the better the reward. And each hill trekked and conquered brings one step closer to Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ. If we recognize this, whatever obstacle is placed on that path, we will find a way to overcome it.
This is similar to how a farmer must be patient to get his crop. He has to labor away many hours working under the hot, dry sun. He has to care for and nurture his garden to eventually reap the reward of his crops. No one said attaining the goal would be easy. But if we change our definition of what easy means to something you can do, then it's easy, "Allah does not charge a soul except [with that within] its capacity. It will have [the consequence of] what [good] it has gained, and it will bear [the consequence of] what [evil] it has earned." (Qur'an 2:286). We trust Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ knows what's best for us as we keep relentlessly marching forward. We turn to Allah, asking for His approval and seeking His guidance, "If Allah should aid you, no one can overcome you; but if He should forsake you, who is there that can aid you after Him? And upon Allah let the believers rely." (Qur'an 3:160)
Abu Hurairah narrated that: The Prophet said: "One of you will be responded to, so long as he is not hasty, saying: 'I supplicated, and I was not responded to.'  Be persistent and do not lose hope. One of the beneficial remedies is persisting in your du'a. Umar Ibn Khattab (ra) said: "I do not concern myself with the response, but I concern myself with making supplication, for I know that the response will come with it!"
The believer further benefits from knowing Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is Al-Mujeeb. Just as Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is responsive to us, we should be responsive to Him. He has asked us to uphold many values, i.e., of kinship, salah, spreading peace, helping the community, and keeping the sawm (fast). The Qur'an tells us (if we're capable) to respond to the ones who ask for help, "and do not chide the one who asks for help;" (Qur'an 93:10). The Prophet ﷺ also never denied an invitation of a person who wanted to help Him even if it seemed unappealing or in this case unappetizing, "Narrated Abu Huraira: The Prophet ﷺ said, "I shall accept the invitation even if I were invited to a meal of a sheep's trotter [feet of sheep], and I shall accept the gift even if it was an arm or a trotter of a sheep." 
 Sahih al-Bukhari 7536
 Sahih Hadith, Jami At-Tirmidhi 3387
 Sahih al-Bukhari 2568