Where does Akbar the Great's "Great" come from?

Jalaluddin Muhammad took the name of Akbar(great) when he was crowned king. A new name/title is not uncommon among the Mughals - Salim took the name Jahangir, Aurangazeb became Alamgir(conqueror), Muazzam took the name Bahadur Shah(brave king), Khurram went by Shah Jahan. Where do the names come from? They just picked whatever name they wanted would be my best bet.

Could not add this as a comment, hence the answer. It would be great to see better explanations to this one.

You are confusing his epithet with his regnal name.

As already mentioned, the practice of Regnal Names was common in the Mughal and some other oriental dynasties. Definition by Wiki:

A regnal name, or reign name, is a name used by some monarchs and popes during their reigns, and used subsequently to refer to them. The term is simply the adjective "regnal", of or relating to a reign, monarch, or kingdom, modifying "name". Since ancient times, monarchs have frequently, but not always, chosen to use a different name from their original name when they accede to the monarchy.

Akbar's birth name was Shahzada (Prince) Jalal ud-din Muhammad. He was given the Kuniyat Abu'l-Fatah. So his fullname became Abu'l Fatah Jalal ud din Muhammad Gorkani (ابو الفتح جلال الدین محمد گورکانی). Moreover, Mughal aristocracy also used the titles Beg (بیگ), Khan (خان) and Mirza (مرزاء) to refer to Royal Princes and in some case, other aristocrats to signify their Persio-Turco-Mongol origins. So formally he would be Shazada Mirza Abu'l Fath Jalal Ud din Muhammad Khan Beg Gorkani.

When Akbar succeeded his father to the throne, he adopted the regnal name Akbar, a tradition that was followed by both his predecessors and successors as mentioned in the earlier answer. The full string of titles that he adopted were:

Original Persian/Urdu

شہنشاہء ھند، السلطان الاعظم و الخاقان المکرم، امام عادل، سلطان الاسلام کفت الانعام، امير المومینین، خلیفتہ المتعلی صاحب الزمان، پادشاہ غازی ظل الہی، عرش آشیانی

Latin Transliteration

Shahanshah-e-Hind, Al-Sultan al-'Azam wal Khaqan al-Mukarram, Imam-i-'Adil, Sultan ul-Islam Kaffatt ul-Anam, Amir ul-Mu'minin, Khalifat ul-Muta'ali Sahib-i-Zaman, Padshah Ghazi Zillu'llah 'Arsh-Ashyani

English Translation

King of Kings of India, the Great Sultan and Exalted Khaqan (Khan of Khans), the Just Leader, Sultan of Islam and source of benevolence, Commander of the Faithful, the Holy Caliph of the Lord of the Age (Prophet Muhammad), Ghazi (A title given to veterans of Jihad) Emperor, Shadow of God (Derived from Hadith that a just King is shadow of God), Dweller of Highest Sky

As you can see, the epithet is not there.

So as far as your questions go that when was he granted the epithet, we don't know. He gave himself the name Akbar But he had taken it as a Regnal name, not an epithet. Akbar means literally the great but that's not the actual epithet you are talking about.

The Epithet here is "Azam", which also means Great/Greater/Greatest. He was referred to by two such epithets:

1. Akbar-e-Azam (اکبرء اعظم) - Which means Akbar the Great
2. Mughal-e-Azam (مغلء اعظم) - Which means the Great Mughal

Later he proved himself through his various military and administrative achievements to be the greatest monrach the House of Timur ever produced, surpassing even his glorious ancestor, Timerlane.

Since these epithets are only to be found in later histories, we can safely assume that Akbar himself never accorded himself the title of being great. It was just a coincidence that he chose himself a regnal name which also means "Great". This is an amusing coincidence, something we would call Ism-ba-Mus'ami (اسم بامسمی) in Urdu, meaning a person whose name fits his character.

For example, see this coin minted during his reign:

If you read the inscriptions, On left side we see the proclamation of Muslim creed:

"La Ilaha il Allah Muhammad ur Rasool Allah" (لا الہ ال اللہ محمد الرسول اللہ)

Translation: There's no God but Allah and Muhammad is the prophet of Allah.

The right hand side bears the name of the Monarch:

Jalal ud Din Muhammad Akbar Ghazi Badishah (جلال الدین محمد اکبر غازی بادشاہ)

Translation: Jalal ud Din Muhammad Akbar, Ghazi King

The epithet "Akbar-e-Azam" is missing from the official coins and also from the official correspondence records made by Akbar, where he just uses his full name and Regnal Name.

It was the later Historians who gave him the epithet "The Great" as there is no mention from accounts of Akbar's own days, where he called himself that. Not to mention, it helps distinguish Akbar the Great from the incompetent Akbar II.