History of zero?
The Wikipedia article 0 (number) is close to the common understanding, and shows there is not a simple claim to invention as there were many early examples of use of zero as a placeholder or number in different civilisations, as well as a philosophical concept. However the points in your debate may have been based based on these two statements:
The rules governing the use of zero appeared for the first time in Brahmagupta's book Brahmasputha Siddhanta (The Opening of the Universe), written in 628 AD.
and
In 976 AD the Persian encyclopedist Muhammad ibn Ahmad al-Khwarizmi, in his Keys of the Sciences, remarked that if, in a calculation, no number appears in the place of tens, then a little circle should be used "to keep the rows". This circle the Arabs called صفر ṣifr, "empty".
with ṣifr being the source of the English word zero (and also cypher).
Disclaimer: most of the dates and historical embellishment in my answer come from the Wikipedia page on the History of the Hindu-Arabic Numeral System.
A full answer requires distinguishing between two different uses for 'zero':
As a placeholder in a positional system of writing numbers. For example, when we write 206 we mean 'two hundreds, no tens and six units'.
As a number in itself, for example in calculations like 5 - 3 - 2 = 0
I'll briefly give some color on these two different uses.
In the first sense of the word, the Babylonians were using a positional system since before 300BC (their system was base 60 instead of the base 10 that we use, but otherwise it is similar). However, they didn't use a zero at the end of a number. This is as if we made no distinction between the numbers 2, 20 and 200 (although they did distinguish between 16 and 106).
The oldest known text using a decimal positional system, in the sense that we understand it today, come from India: the Jain text Lokavibhaga, dated 458 AD (ref: here). The author uses the Sanskrit word for 'void' to denote the positional digit zero.
The first undisputed use of the symbol for zero (0) is an inscription on a copper plate at Gwalior, in India, dated 876 AD.
However, the transmission to Europe of the numeral zero came via the Italian mathematician Fibonacci, who got it from the Arabic mathematician al-Khwarizmi, writing around 976 AD, leading to the common English term 'Arabic numerals' and the belief that the positional system for notating numbers had been invented in the Arab world.
In the second sense, Henry's answer usefully quotes from Wikipedia:
The rules governing the use of zero appeared for the first time in Brahmagupta's book Brahmasputha Siddhanta (The Opening of the Universe), written in 628 AD.
He listed several rules for calculating with zero which would be familiar to use today, including "the sum of zero and a positive number is positive" and "the sum of zero and zero is zero".
However, the word for zero in English again comes from Arabic, rather than Hindi.
This is about the words rather than the concept: I'll address only the etymological origin of the words "zero", "cipher", and "nought" in this answer. (Sources: Online Etymology Dictionary and Oxford English Dictionary.)
zero: circa 1600, (either from Middle Latin zephirum, or French zéro or its source Italian zero, for *zefiro) in any case from Arabic sifr "cipher", translation of Sanskrit śūnya "empty place, desert, naught".
cipher: late 14th century, from Arabic sifr, "zero", literally "empty, nothing", from safara "to be empty", loan-translation of Sanskrit śūnya "empty". (The word "cipher" came to Europe with Arabic numerals. Originally meant "zero", then "any numeral", then (c. 1520s) "coded message". OED: "The Arabic was simply a translation of the Sanskrit name śūnya, literally ‘empty’."
nought: variant of naught which means "nothing". The meaning of "zero, cipher" is only from the early 15th century. (?c1425 Crafte Nombrynge in R. Steele The Earliest Arithmetics in English. (1922) 20: "A 0 is noȝt, And twyes noȝt is but noȝt.")
So these sources seem to agree that:
- In Sanskrit, the word for "empty" (śūnya) was used for zero.
- Correspondingly when translating into Arabic, the word sifr was coined out of the word safara, meaning "to be empty", and used for zero.
- Cipher/*zero* was imported into English, or, similar to the passing from Sanskrit to Arabic, the existing word for "nothing" (nought) was used for zero.