What is wrong with saying "pleased to meet you"?

No, it wasn't because it was a child saying it. It's because in (British) English the 'correct' way to greet someone you have never met before is to say 'How do you do', not 'Pleased to meet you'.

The Daily Mirror has a rather tongue-in-cheek article about how to tell whether you're 'Posh', and using the phrase 'Pleased to meet you' is one of the key indicators that you're not. (Actually, I think reading the Daily Mirror means that you're automatically NOT Posh, but that's just my opinion.)

The terms 'U' (upper class) and 'non-U' (not upper class) were used to differentiate the way the upper class spoke from how the middle class spoke when trying to be 'posh'. Wikipedia (not always a reliable source, I know), gives a list of U/non-U words and phrases. 'Pleased to meet you' is part of the 'non-U' vocabulary, and is (presumably) not something that an upper class person would/should say.

The Telegraph article was from 2010, and the author (Jessica Fellowes) was speaking about her childhood. She was born in 1974 (see Wikipedia), and evidently the U/non-U divide was still going on during her childhood. (She is the niece of Lord Fellowes, who wrote Downton Abbey. It's reasonable to assume that her mother would have wanted her daughter to speak in a way that showed she was a well-brought-up young girl.)

This sort of etiquette is old fashioned these days, and is only likely to be of much use if you are mingling with the upper class and attempting to pass yourself off as one of them (and in Britain, unless you've studied the rules very carefully, you're likely to give yourself away in many different ways before you're even introduced to someone). In general polite conversation, it's absolutely fine to say 'Pleased to meet you'.

The idiomatic expression became inappropriate not because the person who used it was a child, but because of the inappropriate context in which it was used. As per The Telegraph article, the context was: "When I was little, my mother collected me from a school friend’s party. As I clutched my goody-bag and balloon, I shook the hand of my friend’s mother and said: “Thank you. Pleased to meet you” ".

The speaker used it when she was leaving the party, in place of a "goodbye!".

pleased to meet you : (thefreedictionary.com) McGraw-Hill Dictionary of American Idioms and Phrasal Verbs.

an expression said when introduced to someone. Tom: I'm Tom Thomas. Bill: Pleased to meet you. I'm Bill Franklin. John: Have you met Sally Hill? Bill: I don't believe I've had the pleasure. I'm pleased to meet you, Sally. Sally: My pleasure, Bill.

pleased to meet you phrase (en.oxforddictionaries.com)

Said on being introduced to someone.

‘‘This is my wife.’ ‘Pleased to meet you.’’

"Pleased to meet you" is used when meeting someone or being introduced. It could have been phrased, "Pleased to have met you." While stilted, it wouldn't be wrong.

That is, assuming the child met the adult on arriving to the party.

Let's face it. In certain regions of the world, the party host could have been heard saying, "Don't let the screen door hit you on the way out!" It's time to broaden our horizons a bit.

''How do you do'' is LONG gone from almost all public society, and people who would know that the right reply is ''How do you do'' also, are even more rare.

''Pleased to meet you'' is OK. As is ''Pleasure to meet you.'' But 'Nice to meet you' is not very classy at all (although better than ''Hey'' I suppose.

;-)

People may often say ''Good to meet you'' especially in the USA, but ''Great to meet you'' seems more forgivable (as it implies enthusiasm and a more casual setting).

I'm British