I am not going to claim that BTC will. I have no idea whether it will or not. I think it would be great if it did, but I think anyone can only speculate at this point.
IMHO the best argument for price stabilization of BTC is that the liquidity will increase enough, and that the usage will be broad enough. Currently, there are many individuals that can substantially influence the price with their stake. The BTC 'market cap' might be about $5B, but the daily volume is around $10M. That's comparable in marketcap and volume to a stock like SunEdison, and so is the volitility. Currently trade volume is about 20x transaction volume. That's a stock, not a currency.
If the amount traded becomes a smaller amount of the total in circulation, volatility should settle down. That said, even a useful currency like the Euro isn't boring. I tried to find stats on the trade volume of the euro vs. its velocity, but had little success. I'd be curious how it measures up. It would surprise me if the Euro trade volume was 20x it's transaction volume, but who knows.
When it comes to the whole inflation vs. deflation shit, I haven't a clue. One would have thought that QED would lead to some inflation... It doesn't seem that Central Banks can turn the knobs that easily. The fact that hyperinflation exists is pretty good evidence of the psychological nature of currency overwhelming the forces of supply and demand. Brazil had a crazy currency experiment that supports that notion.
At any rate, I don't think it's evident that BTC is going to be a sovereign currency replacement, so it doesn't necessarily have to meet that bar. People are interested in owning plenty of assets that aren't currencies, and to the extent that bitcoin's utility outweighs BTC's shortcomings as an asset, then people should put it to use. For some reason a lot of people are interested in owning paper vouchers for gold. That seems a low bar to me.
There's some crazy interesting shit that is being done with bitcoin that you just can't do with currency. There are decentralized hosting schemes where people that run nodes of a decentralized network get paid out from aggregated BTC to pay for their service. For example, Hubski could be distributed, and if you ran a Hubski node, you would be credited BTC based upon the bandwidth/storage that you contributed. That BTC could come from advertisers, subscriptions, etc., and all the allotment would be automated. You could spin up a Hubski node, and start getting paid within the hour.
There also remains the real possibility that if/when BTC becomes integral to commerce/etc., a central authority will step in and fuck with the code in such a way to make it behave like a Central Bank.