It's a hard one, it depends on how thick-skinned your company is.
I used fanny to describe my bits when I was a bit older. I personally don't think that 'fanny' is offensive, but other people can find it a bit vulgar.
When I was little, we used 'foo-foo'. It's a softer word, slightly nicer to hear if a child says it in company.
"My fanny hurts, Mummy"
"My foo-foo hurts, Mummy"
It's more of a euphemism.
It all comes down to the consonant usage. The double 'N' gives fanny a harsher sound, a bit like a swear-word, so 'CUNT', 'TWAT', 'TITS', 'KNOB', 'DICK', 'FUCK', 'FANNY'.
A word with more vowel sounds is softer, which makes it more socially acceptable because it doesn't sound harsh like an insult, so 'FAIRY', 'FOO-FOO', 'WILLY', 'BOOBY'.