The practice of monks getting a Dharma name upon ordination is very old and widespread.
(It would be interesting to know the flip side question- is there
any Buddhist order than doesn't involve Dharma names?
I can't think of one)
Ref: https://en.wikipedia.org/wiki/Dharma_name
This article discusses the modern practice in Zen of giving Dharma Names where in he mentioned that this is an Indic tradition that probably pre-dates the historical Buddha.
Ref: http://hardcorezen.info/dharma-names/2733
Since the renunciates were renunciating family and householding, it is not surprising that they took on a new name, which makes it perfectly clear to the surrounding community that you don't plan to carry on the Jones family name by having kids, you aren't going to take advantage of the hereditary titles that go with the Jones family nor keep the Jones family estate.
If we're lucky someone familiar with the history of the early vinaya might know more about this, since the practice of Dharma names would have been promoted in the vinaya.
buddhism.stackexchange.com/users/289/MatthewMartin
In India, during the time of the Buddha, the vast majority of monks and nuns didn't change their names.
Notable exceptions are Sariputta and Moggallana , whose original names were "Upatissa" and "Kolita" respectively.
It is not clear why the Buddha chose to call Upatissa by the fact that he was the son (putta) of Sari, or Kolita by the name of his village.
The Mahavagga calls them by these names before they met the Buddha, so it may have been that they took these names earlier.
Other examples that may also have been acquired earlier are Kundala-Kesi, Bakkula, etc.
Cases of adding a descriptive to the name are common, e.g. Kumara Kassapa, Lakuṇṭaka-bhaddiya, etc.
One case stands out, that of Angulimala, who actually had his name changed by the king: Then the king's fear, his terror, his hair-standing-on-end subsided.
He went over to Ven.
Angulimala and said, "Are you really Angulimala, lord?"
"
Yes, great king.
"
"What is your father's clan?
What is your mother's clan?"
"My father is a Gagga, great king, and my mother a Mantani."
"Then may Master Gagga Mantaniputta delight [in staying here].
I will be responsible for your robes, almsfood, lodgings, & medicinal requisites for curing illness."
http://www.accesstoinsight.org/tipitaka/mn/mn.086.than.html Apart from all this, the biggest reason for later name changes is simply because names in other languages don't work for the Pali recitation required for ordination, e.g.: ayaṃ Noaho āyasmato
Tongassa upasampadāpekkho doesn't really work, so giving monks Pali names is considered appropriate: ayaṃ yuttadhammo āyasmato sirimaṅgalassa upasampadāpekkho
It seems that early on there was a tradition of naming monks after arahant disciples of the Buddha, so we get various Moggallanas and Sariputtas throughout the ages.
In modern times, there seems to be some greater variation in the practice, and choosing a name for a monk has taken on significance that pretty clearly wasn't there in the time of the Buddha.
My name, for example, is chosen because I was born on a Wednesday - in fact
, I was given the wrong name, because one monk thought that me being born in the morning in Canada meant that he should give me a Wednesday night name (it having been night in Thailand when I was born).
In the end, What's in a name?
That which we call a rose By any other name would smell as sweet;
So Romeo would, were he not Romeo call'd, Retain that dear perfection which he owes Without that title.
-- William Shakespeare, Romeo and Juliet
buddhism.stackexchange.com/users/18/yuttadhammo
I think this relates to the fact you are starting a new life when you get ordained.
This does not seem to be the case in the Buddha's time but more of a later development.
The name changing practice refer here
is more perticulaly in referece the practice in Sri Lanka where a monk takes the name of the village or area he is from followed by a new name he would be addressed by.
buddhism.stackexchange.com/users/295/Suminda Sirinath S. Dharmasena
It’s worth remembering here that general naming practices in different Asian cultures often differ from our own.
In traditional Thailand (before 1935 or so), government officials received a new name when they took up a new rank in the hierarchy, besides the actual rank title.
For example, there is a scholar of Thai culture who is now known as Phraya Anuman Rajadhon (1888-1969).
Phraya was his official title, the rest is the name he took when he received the title.
(His descendants use A-R as their family name.)
Earlier in his life, he had other names: since his family was Chinese, he was born Li Kuang-jung, and in Thai he was known as Yong Sathienkoset.
When people have multiple names like this, each used in a different context or life phase, another name taken when ordaining would seem perfectly natural.
Note also: in Thailand today, parents seeking a name for a new child will often go to a monk for suggestions.
And a person who wants change his or her given name for luck or some other personal reason will also usually consult a monk.
It may be that this is “not Buddhist,” but anyway it’s common practice.
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I practice with the Triratna Buddhists .
For us the renaming of a person symbolises spiritual death
e.g
the passing away from an old life/way of existence.
Fortunately this is followed by spiritual rebirth or it would be all a bit bleak.
The full process (system of meditation) is Integration Positive Emotion Spiritual Death (new name) Spiritual Rebirth Just responding to @MatthewMartin.
Though it isn't a Buddhist order as such - the Insight Meditation Society doesn't rename as far as I'm aware.
Jack Kornfield is still Jack Kornfield irrespective of his commitment and progress on the path (both formidable I'm sure)
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In the Theravada tradition, and perhaps others, monastics take a new name when becoming bhikkhus/bhikkhunis. Where does this tradition come from, and what is its significance? Others don't - what's the difference?
Why do monastics in certain Buddhist traditions take new names?
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<theravada><mahayana><traditions>
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S0: The practice of monks getting a Dharma name upon ordination is very old and widespread.  I think this relates to the fact you are starting a new life when you get ordained. However, in India, during the time of the Buddha, the vast majority of monks and nuns didn't change their names.  In modern times, there seems to be some greater variation in the practice, and choosing a name for a monk has taken on significance that pretty clearly wasn't there in the time of the Buddha.  Apart from all this, the biggest reason for later name changes is simply because names in other languages don't work for the Pali recitation required for ordination, e.g.:   ayaṃ Noaho āyasmato. An old and widespread tradition, the practice of changing names often symbolises a movement away from an old life or way of existence. This is sometimes referred to as a spiritual death or new life. In India, during the time of the Buddha, the vast majority of monks and nuns didn't change their names. It seems that early on there was a tradition of naming monks after  disciples of the Buddha, with various Moggallanas and Sariputtas throughout the ages. This practice has changed in modern times and the significance of changing a name for a monk is not the same as it was in the time of the Buddha. For example, in Sri Lanka, a monk may take the name of his home village, followed by a new name by which he would be addressed. Names may also have been changed for regional linguistic reasons, because some names did not work in other languages.
An old and widespread tradition, the practice of changing names often symbolises a movement away from an old life or way of existence. This is sometimes referred to as a spiritual death or new life. In India, during the time of the Buddha, the vast majority of monks and nuns didn't change their names. It seems that early on there was a tradition of naming monks after  disciples of the Buddha, with various Moggallanas and Sariputtas throughout the ages. This practice has changed in modern times and the significance of changing a name for a monk is not the same as it was in the time of the Buddha. For example, in Sri Lanka, a monk may take the name of his home village, followed by a new name by which he would be addressed. Names may have been changed because some names did not work in other languages.