Mio may be quick-fingered by certain standards, but I doubt she's anything but ham-handed by standards like those of Itachi. Itachi, one of the two peeps that'd be murdering her, most probably. I rather think their standard is the one that matters here.
Besides trying to shrink down the time it takes to do the hand seals, there's the 'the more familiar with a jutsu you are the less hand signs you need to do' angle. We've flat out seen handseal-less summoning done. The question is... how do you work on that, with a technique that if you perform it you'll remove the utility function that you wanted in the first place? I'm thinking that Storage Seals and other nonliving summoning might be the ticket. There's Tenten and other weapon hoarders, there's Kotetsu and Izumo with that hammer-shell thing - something might work there, for a bullshit Uchiha genius to use to crib experience with a trick she dare not yet use.
Past making the hand seals faster and reducing the number needed, there's a third angle to approach the impending problem; ensuring Mio
survives long enough to apply said skill. After all, no matter how fast she may get, I doubt we'll be surprising Itachi
or Obito with it, not without some luck. Rather than rely on that, I'd rather like to look at the well established field of 'staying alive when you shouldn't be/healing things that should be killing you' field that is Medical Jutsu. The trick here, is that Danzo's going to be the one giving Itachi the orders - and with his obsessive use of Root, there's no chance Mio could go learn from hospital peeps
without that then being listed on her dossier when it's handed to Itachi alongside everyone else's. Somewhere between 'Hiding actual ability in official lessons to the point you don't seem like you're making the cut' and 'finding a sufficiently sneaky and innocuous roundabout route to getting medical training', I think there's an answer. It might involve making friends with at least
a Hyuuga - I imagine Root's got standing orders regarding how to avoid detection around them, and that just might be enough to let some genin hide their actions.
So in short; if we want to do all we can to ensure Mio survives THE MASSACRE, she needs to:
- Work for finger dexterity and general handspeed of the sort that Itachi would be impressed by (while well aware we almost certainly aren't gonna hit that mark).
- Dabble in all matters summoning besides the official contracty sort to get as familiar with the method as possible.
- Pick up Medical Jutsu in a fashion that doesn't alert whatever ROOT assholes are on rotation to watch a semi-notable Uchiha.
- (Potentially) Make use of Hyuuga Byakugan-ness as a ROOT-away system of sorts.
- (Potentially) Hang around notable and cool-guy Jounin that both could detect such watchers, and dislike their presence.
That... actually doesn't seem all that challenging as a to-do list, nor does it look like the kind of stuff that'd chew up quest update space if one tried to implement it; nice!