Of Australian-English origin, the noun babyccino (also babycino and babychino) is from the noun baby and ‑ccino in the noun cappuccino, and designates:
– a children’s drink consisting of hot milk that has been frothed up with pressurised steam so as to resemble cappuccino;
– a small cup of cappuccino—cf., below, quotation 2.

The noun cappuccino designates a type of coffee made with espresso and milk that has been frothed up with pressurised steam—cf. The literal meaning of ‘cappuccino’ is ‘Capuchin’ .

The noun babycino, for example, in the following from A child in time, by Helen Brown, published in The Daily Telegraph (London, England) of Saturday 20^th July 2013 [No. 49,189; Review section: page 29, column 4]:

The cutesy adaptation of adult culture for small children should be treated with deep suspicion. Cafés flogging thimblefuls of warm milk, priced as “babycinos”, peddle the fantasy that you can sit down and enjoy a relaxed, cosmopolitan break with a toddler when in fact you will inevitably end up scraping foam from your hair while your pudgy companion gets high on the contents of the sugar bowl.

The earliest occurrences of the noun babyccino (also babycino and babychino) that I have found are as follows, in chronological order:

1-: From The Sunday Age (Melbourne, Victoria, Australia) of Sunday 14^th May 1995 [Life! section: page 3, columns 1 to 4]:

The no-coffee coffee set
First, there was cappuccino, then de-caf cappuccino and caffe latte. Now it’s babyccino, the no-coffee coffee for the junior set. Anne Crawford reports.
[…]
The babyccino is sweeping Melbourne’s kindergarten set as the acceptable alternative to apple juice for cafe-sitters in training wheels. Eateries in Brunswick and Acland streets, Carlton’s Italian cafes, St Kilda’s Stokehouse (downstairs) and the Continental Cafe in Prahran, all serve them.
It has joined the double de-caf, the Skinny cappuccino, and any number of trendy variants on the everyday cappuccino immortalised by Steve Martin in the “coffee scene” in ‘LA Story’. The only difference is that it doesn’t contain coffee.
Also called the “albinoccino” or the “bubaccino”, the beverage is mostly milk froth with a dusting of chocolate and the option of a submerged marshmallow. Usually costing 50 cents—although sometimes complimentary for regulars—the babyccino is rarely, if ever, advertised. It’s up to parents to know where to get one.
[…]
But babyccinos are not just for juniors. Sails on the Bay restaurant in Elwood has noted at least one adult customer who asked for a “cappuccino—no chocolate, no coffee”.

2-: From the column Upfront, edited by Cassie McCullagh, published in The Sydney Morning Herald (Sydney, New South Wales, Australia) of Saturday 18^th November 1995 [No. 49,364; Good Weekend section: page 13, column 1]:

Gone are the days when a latte or a macchiato was the fanciest coffee available. New cafes seem to spring up in every vacated shop, and with them come clever variations on your standard cup-of-cino. […]
[…]
Espresso Con Panna—the Kylie Minogue ¹ of coffee
A short black with a dollop of cream on top. Appears sweet, but is most definitely bitter underneath.
[…]
Babyccino—the John Howard ² of coffee
A short black-sized cappuccino. Just a small version of the real thing.

¹ Kylie Minogue (born 1968), Australian singer and actress.
² John Howard (born 1939), Prime Minister of Australia from 1996 to 2007.

3-: From the Travel pages of The Sunday Age (Melbourne, Victoria, Australia) of Sunday 25^th August 1996 [Agenda section: page 12, column 6]:

Noosa Heads
If you think Melbourne’s perfect except for the weather, Noosa is your town. It has great restaurants, classy shopping and you can get a good cup of coffee. In many ways it’s the Lorne of the north. The Sheraton was one of the first resorts in Australia to have a creche, and the Hastings Street coffee shops probably invented the “babycino”.

4-: From I ask you, by Jenny Tabakoff, published in The Sydney Morning Herald (Sydney, New South Wales, Australia) of Saturday 26^th October 1996 [No. 49,657; Spectrum section: page 2, column 2]:

Often, you find yourself asking questions that reveal rather more about yourself than you would like. “Do you serve babyccinos?”, “Which kid is yours?” and “Can you tell me where the parents’ room is?” spring to mind.