Sure, they'll still be collected as vintage items for their sentimental value the way old NES games are still collected. But mining them will be easy enough that they'll have no real value.
How would that be, given that mining becomes more and more difficult over time by design?
less ppl mining = difficulty drops
Maybe I should have qualified that with "and there is a limited supply by design". At this point in the game, I can't foresee us reaching a point where there would be a significant number of coins left over that would be "easy enough" to mine, regardless of the dip in value. With the impending number of people with ASIC hardware, at a relatively small electricity cost, they will be able to mine the vast majority of the remaining bitcoins by 2033. As improvements happen with ASICs, maybe the price comes down on them, or maybe they maintain a premium cost but with increasing hashrates and lower power requirements. It just seems more likely that there would be a select set of enthusiasts doing the mining for the long term and it wouldn't ever be "inexpensive" for just anyone to jump in and mine a significant number of coins for themselves. Of course, I could be wrong.
Seeing that we have ASICS up and running, the difficulty is never going to get back down to CPU profitable levels. There are bitcoin enthusiasts who will mine at a loss just to keep the network going. But if bitcoin prices go way down and usage drops, you will be able to pick up a used ASIC for pretty cheap, and so mining could turn out to be cheap after all. The other (IMHO, more probable) extreme is that prices and usage go up, and so mining will continue to be expensive, always trending toward zero profit.
Remember, the miners get to keep the transaction fees, so in the future even after all the bitcoins are minted there will still be some reward for miners. But transaction fees are dependent on usage, so if nobody is using bitcoins then mining will have no reward.