The original poster asks three questions about Mary approaching Jesus regarding the wine situation at Cana. Many interesting answers have been posted and I have only my own opinion to contribute.
Let me give my answer to the second question first:
"Was the family very close to her/them ?"
John gives us a hint about the relationship of the families:
' On the third day there was a wedding in Cana in Galilee, and the mother of Jesus was there.
Jesus and his disciples were also invited to the wedding.'
The obvious naive interpretation here is that Mary was a friend of the bride or the groom or their families or some combination of this, and that Jesus and his disciples were invited as a courtesy. I think she was most likely a friend of the mother of the bride as she has a roughly thirty year old son and the ages work out, and a female friend seems more likely than a male. Possibly a relative also, look at how many people of note in the Gospels are related, e.g. John the Baptist is Jesus' cousin and so on.
to amplify: If I wrote about a wedding in the early US, "Abigail Adams was there, and John Quincy Adams and his friends were also invited", wouldn't one's natural assumption be that Abigail is the primary guest and the others mentioned were invited because of her?
(Actually some of the other posters have inspired me to speculate what this might 'mean' for humanity if true - I think that the wedding feast might be like our little world, and that Jesus came here for Mary's sake, that without her Jesus would be doing other things. I think God has /plenty/ of worlds, universes, to take care of, and that if anything made him single out this world for special attention, it was Mary. The rest of us are just 'baggage', what God wanted from our universe was Mary - he conceived her as a perfect character (immaculately!) then I think the rest of our world is arranged around her.)
Now my answer to the first question:
"Is there any indication or insight as to why she did so ?"
John tells us
'When the wine ran short, the mother of Jesus said to him, “They have no wine.”
I see Mary as a close friend of the family pointing out a problem to Jesus, she is pointing it out BECAUSE it is her friends who have the problem. It's nothing to do with him in some sense but she is concerned for her friends. In particular the parents who are providing the refreshments might look bad and I think Mary felt for them.
[And] Jesus said to her, “Woman, how does your concern affect me? My hour has not yet come.”
Jesus seems not to understand Mary's concern here. Now we are pretty sure he really does understand the problem, even if he weren't God he was still very intelligent - we know he confounded educated people at the temple, that he seemed like a prodigy to them when he was much younger. Mary knew him better than anyone, much better than we do, she knows he doesn't mean "I don't understand the situation." I read this as a much more sophisticated exchange. We know he is an adult at this time and even if she is humanity's greatest genius (which I suspect she might be) he is something more than that. So my guess is that he is teaching her something, if she wants to learn it, and incidentally, teaching us something as well.
What could he be teaching her? or us? I don't know but what I see here is that he is saying that he is detached from the situation, that the time hasn't come for him to interfere in human affairs. I'm reminded of stories about ascended beings in science fiction and the like. Jesus could solve this problem but he could solve /any/ human problem if what we (Catholics at least) believe about him is true - he could have the Romans leave Judea, he could make humans immortal, he could bring back unicorns and so on. But he doesn't work that way. People are free to do what they please and have to accept the consequences of their choices. It isn't time for him to fiddle things around and fix things, without being invited.
And that brings us to this:
His mother said to the servers, “Do whatever he tells you."'
She doesn't tell Jesus "They have a problem, fix it." She doesn't say "make a miracle". If the childhood legends of Jesus are true she knows he can do miracles, certainly she knows he is special from what Gabriel told her! Plus he grew up with her, even if she has forgotten he is miraculous or divine, she knows he is a genius, he can figure out a solution that will save the day.
But he says 'My hour is not yet come.' I don't think his hour is set on a schedule, at least not one that she knows! It's sometime in the future, and she figures it out, because she is a genius also! Not surprising that she is no dummy.
His hour comes when he it comes. She causes it to come!
She tells the servants, "Do what he tells you." And she sets the ball rolling for the rest of the story, his rise to fame and glory. She knows this is risky, but it is she who triggers his hour coming by telling the servants to obey him.
What can we learn from this? Again, I don't know, but to me the obvious thing is that Jesus leaves us to live life without interference until he is invited. Christians are like the servants whom Mary tells, "Do what he tells you" (she says something similar elsewhere to people in general but I haven't found it just now). Our job is to make the wedding nice for everyone, including non Christians, we are just the servants. Jesus can handle all the miraculous parts and can tell us the strategy but he allows us to help by fetching water to fill the jars and so on - he tells them to fill the jars with about 100 gallons of water, note that he doesn't just miracle that water in there. So there is plenty for us to do. I also think that whatever we do actually helps, sure He doesn't /need/ the help, he could just miracle away problems or fetch the water himself, but that's not the /optimal solution/, the optimal solutions involves us doing some of the work. I think that miraculous solutions are probably /just as much work/ as normal methods, or /more/, but since God does all the heavy lifting, we say "Oh that looks easy".
"Or was it simply the time for Jesus to perform a public miracle and it came about in this manner?"
I don't see this but it sounds insightful. I feel it worked the other way around, that this was the first public miracle because of the situation, and that if it was "time for it" that just means that the overall storyline of the universe had a miracle-shaped space there. Hmm this makes a bit of sense. But my feeling is that we can study the actual storyline and learn more than by studying some screenplay formula that says "miracle at the end of act two" in general, which what I take "time for it" to mean. Apologies if I am missing the point.