Apparently so, Silvio. Here's a link explaining the general rules of Italian imperatives, and a few exceptions: http://italian.about.com/library/weekly/aa011900a.htm
El, the imperative of venire is highly irregular. In fact nearly all the tenses of venire are highly irregular. See the bottom of
They are both translated as "here", just like lì and là are both translated as "there". But there are some fine nuances which we can't express in English. Quì is more precise than quà. You could translate quì as "exactly here in this spot", and quà as "around here". Same rule applies to lì and là, with lì being the locally more precise one. I'm not a native Italian speaker though, so take what I wrote cum grano salis.
'Vieni qui' is the informal you 'tu' imperative, 'Venga qui' is the formal 'Lei' imperative. See the imperativo section at the bottom of http://www.wordreference.com/conj/ITverbs.aspx?v=venire.
She is, of course, correct to use the 'tu' form with you since you are family. The 'Lei' form would be used with someone to show courtesy or respect, for example, a customer at a shop where you work. Took me a while to figure it out, but the 'Lei' imperativo is the same as the congiuntivo presente, which makes it easy to remember.
Yep, that's confusing!
I am completely sure that 'venga qui' means come here. Since I'm no expert, that's probably not a lot of help.
However: translation sites can be unreliable - language is too context-driven for them to wrap their little binary heads around. Individual words are usually cool, whole phrases less so. They are getting better, but need to be used with caution at the moment.
My personal go-to for looking up a phrase is CONTEXT REVERSO. It's a massive, searchable database of translated texts. Many are from TV show and movie subtitles, so don't be surprised when you see Spock or Starbuck mentioned, but they also include formal documents. You enter the phrase you're interested in, and it will give you a statistical break-down of the different ways it was translated in the past. It's like using human expert brains as a multiple search engine. Really, really useful. I can't recommend it enough.
You'll see 'venga qui' there, no problem.
Have a sunshine day. (Winter here ... freezing my assets off.)