Arabic Root:
From the root ayn-dal-lam (ع د ل), which has the following classical Arabic connotations: to act justly, fairly, equitably, to adjust properly, to make even, to straighten, to rectify, to establish justice or balance. To make equal, uniform, to turn one away from something, rightly, direct, to make comfortable with what is right.

Among the disputed names:
We've mentioned previously how varying scholars have different criteria for what constitutes being a name of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ. The name Al-Adl is not recognized as an official name by scholars such as Ibn Uthaymeen, Ibn Hazm, and Ibn al-Wazir, among others. As a rule of thumb, if the name contains the prefix alif lam before the name in the Qur'an, then that's a clear indication of belonging as a name of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ. There is no verse directly quoting Al-Adl in the Qur'an.

In Sahih Muslim, “O My servants! I have forbidden dhulm (oppression) for Myself, and I have made it forbidden amongst you, so do not oppress one another.” Here Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ makes a declaration, He can do anything He pleases, but He forbids injustice for Himself. A point made to let us know that He is not unjust. Allah then commands us to uphold this standard amongst ourselves, to call out and prevent any injustice we may see.

إِنَّ ٱللَّهَ لَا يَظْلِمُ مِثْقَالَ ذَرَّةٍۢ ۖ وَإِن تَكُ حَسَنَةًۭ يُضَـٰعِفْهَا وَيُؤْتِ مِن لَّدُنْهُ أَجْرًا عَظِيمًۭا

Innal laaha laa yazlimu misqaala zarratinw wa in taku hasanatany yudaa’ifhaa wa yu’ti mil ladunhu ajran ‘azeemaa

English Translation:
"Indeed, Allah does not do injustice, [even] as much as an atom's weight; while if there is a good deed, He multiplies it and gives from Himself a great reward."— (Qur'an 4:40)

إِنَّ ٱللَّهَ يَأْمُرُ بِٱلْعَدْلِ وَٱلْإِحْسَـٰنِ وَإِيتَآئِ ذِى ٱلْقُرْبَىٰ وَيَنْهَىٰ عَنِ ٱلْفَحْشَآءِ وَٱلْمُنكَرِ وَٱلْبَغْىِ ۚ يَعِظُكُمْ لَعَلَّكُمْ تَذَكَّرُونَ

Innal laaha ya’muru bil ‘adli wal ihsaani wa eetaaa’i zil qurbaa wa yanhaa ‘anil fahshaaa’i wal munkari walbagh-i’ ya’izukum la’allakum tazakkaroon

English Translation:
"Indeed, Allah orders justice and good conduct and giving to relatives and forbids immorality and bad conduct and oppression. He admonishes you that perhaps you will be reminded."— (Qur'an 16:90)

وَتَمَّتْ كَلِمَتُ رَبِّكَ صِدْقًۭا وَعَدْلًۭا ۚ لَّا مُبَدِّلَ لِكَلِمَـٰتِهِۦ ۚ وَهُوَ ٱلسَّمِيعُ ٱلْعَلِيمُ

Wa tammat Kalimatu Rabbika sidqanw wa ‘adlaa; laa mubaddila li Kalimaatih; wa Huwas Samee’ul ‘Aleem

English Translation:
" And the word of your Lord has been fulfilled in truth and in justice. None can alter His words, and He is the Hearing, the Knowing."— (Qur'an 6:115)

Why is there suffering?
The topic of suffering can bring about doubt in a believer's heart. The uneasiness we feel is because we can't comprehend Allah's justice. We may have uncertainties, fear, or forget this world is only but a temporary residence. We must remember everything is within Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ ownership and control. He can do whatever, with whomever, and He is the All-Wise and All-Knowing. If we understand this, we don't need to go any further. Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ can dispose of all of creation today, and it wouldn't be right or wrong; it's just His command. This is what it means to be the Supreme Authority, King of Kings, and the All-Powerful. To deny this would be to deny His attributes. But only seeing Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ only through the lens of His independence and sovereignty misses His essence. It is more comprehensive seeing all His attributes working together, including Him being the All-Knower, the All-Wise, The Merciful, and The Embodiment of Justice.

Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ tells mankind, "Surely We will test you with a bit of fear and hunger, and loss in wealth and lives and fruits, and give good tidings to the patient." ( Qur'an 2:155) And Prophet Muhammad ﷺ said, “No fatigue, nor disease, nor sorrow, nor sadness, nor hurt, nor distress befalls a Muslim, even if it were the prick he receives from a thorn, but that Allah expiates some of his sins for that.” So, we must remember this life is only temporary, and with "every hardship is ease," whether it is realized in this life or the hereafter. This is Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ justice - it will always be fair and measured. "Allah does not task any self beyond its capacity" (Qur'an 2:286).

On the suffering of a child:
For every child that passes away, they are granted eternal bliss in Paradise. In a hadith, it reports Allah's Messenger (ﷺ) who used to ask his companions, "Did anyone of you see a dream?" and they told him their dream. He commented on this dream, " And the tall man whom you saw in the garden, is Abraham and the children around him are those children who die with Al-Fitra (the Islamic Faith). Some Muslims asked the Prophet, "O Allah's Messenger (ﷺ)! What about the children of pagans?" The Prophet (ﷺ) replied, “And also the children of pagans.”

Abu Hassan reported: I said to Abu Huraira that my two children had died. Would you narrate to me anything from Allah's Messenger (ﷺ) a hadith that would soothe our hearts in our bereavements? He said: Yes. Small children are the fowls of Paradise. If one of them meets his father (or he said his parents) he would take hold of his cloth, or he said with his hand as I take hold of the hem of your cloth (with my hand). And he (the child) would not take off (his hand) from it until Allah causes his father to enter Paradise.

Abdullah ibn Amr reported: The Messenger of Allah, peace and blessings be upon him, said, “Verily, those who were just will be in the presence of Allah upon pulpits of light, near the right hand of the Merciful, the Exalted, and both of His sides are honorable. They are those who practiced justice in their judgments and with their families and in all that they did.”

۞يَـٰٓأَيُّهَا ٱلَّذِينَ ءَامَنُواْ كُونُواْ قَوَّـٰمِينَ بِٱلۡقِسۡطِ شُهَدَآءَ لِلَّهِ وَلَوۡ عَلَىٰٓ أَنفُسِكُمۡ أَوِ ٱلۡوَٰلِدَيۡنِ وَٱلۡأَقۡرَبِينَۚ إِن يَكُنۡ غَنِيًّا أَوۡ فَقِيرٗا فَٱللَّهُ أَوۡلَىٰ بِهِمَاۖ فَلَا تَتَّبِعُواْ ٱلۡهَوَىٰٓ أَن تَعۡدِلُواْۚ وَإِن تَلۡوُۥٓاْ أَوۡ تُعۡرِضُواْ فَإِنَّ ٱللَّهَ كَانَ بِمَا تَعۡمَلُونَ خَبِيرٗا

Yaa aiyuhal lazeena aamanoo koonoo qawwa ameena bilqisti shuhadaaa’a lillaahi wa law ‘alaa anfusikum awil waalidaini wal aqrabeen iny yakun ghaniyyan aw faqeeran fallaahu awlaa bihimaa falaaa tattabi’ul hawaaa an ta’diloo; wa in talwooo aw tu’ridoo fa innal laaha kaana bimaa ta’maloona Khabeera

English Translation:
"O you who have believed, be persistently standing firm in justice, witnesses for Allah, even if it be against yourselves or parents and relatives. Whether one is rich or poor, Allah is more worthy of both. So follow not [personal] inclination, lest you not be just. And if you distort [your testimony] or refuse [to give it], then indeed Allah is ever, with what you do, Acquainted."— ( Qur'an 4:135)

Reflection:
In recognizing Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ as just, how can we bring this virtue into our life? Instead of seeking confirmation or looking to prove our side as being correct, can we instead seek the truth? This would necessitate leaving our emotions and ego at the door. We would have to embrace our weaknesses and the possibility of being wrong. We can also expand the scope of justice to go beyond interpersonal dynamics. We can look within the individual. Are we just to our souls, treating ourselves with respect, or are we oppressive to ourselves?

References:
01) Sahih Muslim 2577a
02) Sahih al-Bukhari 5641
03) Sahih al-Bukhari 7047
04) Sahih Muslim 2635
05) Sahih (authentic) Sahih Muslim 4493