Several of the SHs and their friends have missing pieces of their backgrounds. I'm only hitting the major characters here, or characters likely to become big characters:
STRAW HATS:
1. Luffy: The more we learn about Luffy, the less we know. We know he had a grandfather (a mysterious entity in and of himself), an older brother, and that's it. But just the appearance of Ace and the mention of "Grandpa" led to tons and tons of questions about Luffy's past, especially who his grandfather is and where Ace was during Romance Dawn.
2. Zoro: We know ZERO about Zoro's family - the VIZ manga doesn't even list hold old he was when Kuina died or when he joined the dojo. The anime has him running around challenging dojos, but this is most likely just filler.
3. Nami: We know a LOT about Nami, and the likelihood of us ever finding out who her birth parents were is zilch. Even if we did, it was Bellemere who raised her and that's who she sees as her mother.
4. Usopp: We know quite a bit about Usopp, too. We know who both of his parents are and roughly what happened to each of them. He spent his childhood alone running through the streets of Syrup village shouting about pirates and then going on to form the Usopp Pirates. The more interesting piece of Usopp's past will be when he meets with his father.
5. Sanji: Sanji was nine or ten when the Orbit sank and Zeff took him in. We also know Sanji came from North Blue and was in a steady enough home life to be read stories such as Liar Norland's. This makes me wonder how Sanji ended up on the Orbit (and for that matter, in East Blue) if he clearly had some semblence of a family at one point.
6. Chopper: We know everything from birth to now. Shunned by the reindeer, eats the Human Human fruit, shunned by the reindeer AND the humans, taken in by Hiruluk, ect.
7. Robin: We know enough about her past right now, as the Ohara flashback covered a large span of her life, the earliest being when she was two and the latest being when she joined Crocodile.
OTHERS:
1. Ace: Well duh. Where was he during Romance Dawn? I'd also like to see a flashback to the day when HE left his village and learn how he got involved with Whitebeard's crew.
2. Franky: I know he's PROBABLY going to be a Straw Hat, but as of this writing only chapter 420 has been released so it's not official yet. Young Franky told Tom he was abandoned when he was fairly young - it seems slightly likely to me that we're going to find out more about that situation in the future.
3. Shanks & crew: What have they been up to, aside from sending a message to Whitebeard, for the past ten years? I'm also curious about Yassopp's recruitment and whether he joined them before or after Usopp was born.
I'm guessing that most people here don't know the answers to these questions (as most weren't addressed by the manga in detail yet), but I thought it might be interesting to speculate on. Any takers?