nice question :)
normally we use names. I'm Francesco...so my dad calls me Fra, or Checco, or Franci..

In a bit elegant way we use: Caro, cara (dear)

In an overattached way, mostly mothers say: Amore mio (my love) when the child is still very young, otherwise it's weird.

Often, always mothers, say "tesoro" (kinda like honey).

All these words are also used to call you lover! Piccolo or Piccola today in italy are mostly romantic, but used also for babies or little children, at least from my experience.

<em>
</em>

<em>Francesco</em>

I have two children and when they were babies I usually called them "patatina" and "patatino". weird isnt't it?  but little potato in english doesn't sound very romantic!

my mum usually calls me "topolina" or "dolcezza", but I think the most used is "amore" :)

Grazie a tutti!

We called those nouns "vezzeggiativi" like:

orso=orsETTO

piccolo=piccolINO

labbro=labbrUCCIO

I think that if you seach on google: "vezzegiattivi italiani" you will find a good and complete explaination.