(Inspired by this euphoric fever dream of an infographic.)
There will continue to be nothing that Bitcoin does better than existing systems, apart from money laundering and purchasing illicit goods.
No, not remittances. The expensive bit of Western Union is in fact that last mile.
It will remain difficult to turn your Bitcoins into conventional currency (which is the only reason there's such a spread between exchanges).
It will get even harder to turn your conventional currency into Bitcoins, as any exchange not being run by blatant crooks puts you through the anti-money-laundering mill.
The protocol problems will continue not to be fixed, unless most of the hashing power and Mircea "socks and cocks"* Popescu can be convinced to go along with the Bitcoin Foundation. No 20-meg blocks for you!
99% of current hashing power came online in 2014; this will be very price-sensitive, and much will go offline as the price drops, maybe coming back next hash adjustment.
Miners will continue to sell their coins immediately to cover costs: we are circling equilibrium, where the cost of mining 1 BTC is about 1 BTC. The pool of money to pay for them comes from new Greater Fools.
Transaction irreversibility will remain Bitcoin's sticking point, as speculators who are insufficiently computer-savvy keep getting burnt. "No chargebacks" will continue to repel customers and not attract businesses.
Everyone who bought in the last year and held is a bagholder. Their claims and speculation will get increasingly frenzied. Ask for numbers supporting all claims, particularly the ones in the above-linked infographic.
The bagholders and gambling addicts will continue to be taken by obvious scams, e.g. the two Ponzi scheme sites in just the last month.
Sidechains will continue to be vaporware and not in fact a thing that exists, let alone solves any problems. Bitcoiners will still talk about them as if they exist in the present, therefore you should ignore that altcoins are possible. Edit: OTOH, pettycoin might get finished this year.
Altcoins will continue to be even scammier than the Bitcoin ecosphere, boggled as I was to realise this.
The price is presently being held up by speculation and wishful thinking. No new reason will come along. The "fundamentals" are a castle in the air.
Nobody actually wants smart contracts. They know that the plot of Dr. Strangelove is literally an unstoppable smart contract going wrong. Real customers want problems to be fixable when circumstances change, real companies want to retain the option of lawyering out of a stupid deal. The only people who would want smart contracts are businesses looking to screw over their customers even more than "mandatory arbitration" clauses do. This is about as appealing to customers as no chargebacks, for the same reason.
Blockchains, even if by some remarkable wrinkle they turn out useful for something, will not lug Bitcoiners' 33 GB of SatoshiDice penny shavings with them. Bitcoiners will continue to bring up "blockchain technologies" as a reason to bother with Bitcoin regardless, because that's literally all they have.
(Bitcoiners misunderstand that when a techie calls something "interesting" they don't necessarily mean "useful", "feasible" or "practical" — often they mean "what the hell even is that" or "I ain't even mad, that's amazing". The blockchain, particularly as implemented in Bitcoin, is very much the last.)
If you know nothing about Bitcoin and find the above largely confusing, here's the short FAQ and the RationalWiki article (which I started). I'm not such a fan.
Update 2016: Pettycoin still isn't finished. Otherwise, pretty good.
* Technically this is blatant ad hominem, but it's definitely a post so amazing it should be linked anywhere his name is mentioned, ever. Archive link.