I think "puns" is a correct translation, and you should report it if it isn't accepted already.

giochi di parole is italian for pun, but the english phrase 'word games' encompasses much more than puns.

I also wrote "she likes puns" and it was marked incorrect. I don't know if you ever reported it, but if you did, two years later it is still not accepted. I reported it- 9/28/2016

The translation of "they like word games" is "a loro piacciono i giochi di parole". You can also say "gli piacciono" but this can mean both "they like" and "he likes".

(American English speaker) "la" would be a direct object. Here it is "to her"

'Piacere' can be roughly translated to 'to be pleasing' - if 'she likes the word game', this means 'the word game is pleasing to her', which in Italian is 'Il gioco di parole piace a lei.' But generally, having the indirect pronoun come first is preferred (a lei - le), giving 'to her, is pleasing, the word game'. In Italian, this is 'Le piace il gioco di parole'.

If 'she likes the word games', this actually means 'the word games are pleasing to her' (I giochi di parole piacciono a lei), or reordered, 'to her, are pleasing, the word games'. The subject, most importantly, is the 'word games' (in reality, it is they that are pleasing her, and not she who is 'liking' the word games). This means that, as in the English are pleasing, the verb needs to be in plural form, 'piacciono', giving 'Le piacciono i giochi di parole'.

"-cere" verbs usually have a spelling change, where any form that has an "io" or "ia" (first person singular "piaccio", first person plural "piacciamo", third person plural "piacciono") doubles the c.

'Le' means 'to her' 'gli' means 'to him'. "He likes word games," would be "Gli piacciono i giochi di parole."