edit: I feel this is bit relevant: http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_m...
Maybe. I'm not sure I want my calculator showing an alert about what my friends are doing tonight when I'm trying to focus.
The cool part is when you start thinking about how to innovate on top of that. Then you can add things like:
* the ability to display and remember the results of a whole range of calculations on a single screen instead of always having to write stuff down or bring back old calculations, e.g. http://stuff.stdout.be/graphcalc.png (think iPad app)
* the ability for teachers to follow along on their own screen as you're doing calculations, to help out, maybe even allow teachers to take over control (remotely) to show stuff
* save calculations and make little reports, which could serve as homework
* slowly learn general-purpose programming along the way, simply by having the thing based off a language like CoffeeScript instead of Texas Instruments' own half-assed TI-BASIC.
* extensions for physics and chemistry, so it's not just about math
I don't know if it'd really make sense as a startup (tablets are still too expensive and the nice thing about graphing calculators is that they're small and people can't browse the internet on them) but the whole "let's try to reinvent some ancient technology using modern tools" thing is pretty fun.
If anybody else is looking for a weekend hack project, I'd gladly open-source what I have thus far.
Replicating the interface is key. From my experience, most teachers were horribly confused if someone brought in a TI-89 or, forbid, a TI-92. Why? Different UI that they don't know. You can't teach students how to use their calculator unless you know how to use it yourself.
After, you can add in all sorts of extra features like you mentioned. But "A TI-83 that runs on iOS" is probably the first feature that people will pay for.
It does bring up the issue of students cheating on tests, using their iOS "calculators", but last time I checked it's not too hard to hide notes on your TI-83 either...
It's an idea I've had for an iOS app for a while, but don't have the time to dedicate to it (working on something else).