Let's find some sort of intrinsic value that BTC has (besides scarcity or BTC's usefulness as a medium of exchange). This will put a price floor under BTC.
To get things started, I thought of a few. I'll copy them here for easy reading, but please post replies to this at https://en.bitcoin.it/wiki/Intrinsic_worth_brainstorming
=== BitX ===
. The worth of bitcoins would be the maximum of the worth of any of the services built on top of bitcoin.
Many of the following ideas are a specialization of the BitX idea.
=== Publishing ===
Bitcoin is fundamentally a global censorship-resistant memory system, similar to FreeNet. It just happens that what it is remembering is the log of Bitcoin transactions.
Information can be encoded in the transaction log, and the speed with which it can be encoded is limited by the ability to encode information into the blockchain. Encoding information into the blockchain requires BTC (or, while BTC are mineable, computing power, which has a price floor).
Publishing services could be setup which encode information into the blockchain (that is, to perpetually, globally publish, in a censorship-resistant way), and read out encoded information, for a price. Like FreeNet, for pay. This sort of service is more valuable if it is done on top of something like Bitcoin (because the network effects of Bitcoin provide stronger incorruptibility). This inherent value provides backing.
=== Marked coins ===
Because BTC is traceable, other alternative currency systems could be built on top of BTC. For example, if I had a lot of gold, and assuming that 1 BTC was worth less than an ounce of gold, then I could buy 1000 BTC and then say, "My BTC address is XYZ. Any BTC which was has passed thru this address is equivalent to a transferrable IOU for one ounce of gold from me.". Then I sell those BTC for whatever an ounce of gold is worth. This provides a standardized way to electronically transfer gold IOUs.
This would mean that those bitcoins would be backed by gold. It wouldn't mean that ALL bitcoins would be backed by gold. But, the fact that all alternative currency issuers utilizing this "marked coins" scheme on top of bitcoin first need to acquire bitcoins provides a backing for all bitcoins (the value of this backing would be much less than the value of the alternative currencies themselves, of course). In other words, bitcoin's inherent usefulness as a layer underlying other alternative currencies would be the backing.
=== Namecoin ===https://bitcointalk.org/index.php?topic=6017.0
=== Goods and services which are ONLY buyable with BTC, for set prices ===
Mere usage of BTC as a medium of exchange isn't backing. But if some merchants sell something which (a) may ONLY be purchased with BTC (similar to "petrodollars"; a lot of oil can only be purchased with dollars), (b) are made available to sell at a set BTC price, INDEPENDENT OF THE EXCHANGE RATE OF BTC VS ANY OTHER CURRENCY, then this provides backing.
An example would be an MMORPG in which the in-game currency were Bitcoin, and in which the prices of some in-game things were set by the creator of the game without reference to BTC's current price on currency exchanges (specifically, if the price of BTC in dollars fell, the game maker would NOT drop the in-game price of items).
Something which satisfied (a) but not (b) wouldn't quite be backing, but it would be better than nothing.
=== A club which demands dues in BTC ===
Consider how the U.S. dollar is the only currency that is accepted for payment of U.S. taxes. This creates a demand for dollars to pay taxes with.
The U.S. gov allows you to transact business in some other currency, but you still have to pay a percentage of the net income generated by those transactions in dollars.
This isn't quite backing but it would be better than nothing.
== Alternatives to BTC which would have intrinsic worth ==
This section is for discussing things that can't be done ON TOP of BTC, but which could be done by creating something very similar to BTC.
=== pseudo-BTC that uses computational power in service of some problem ===
If there were a way for someone to purchase a large amount of pseudo-BTC and then could use them to "buy" computing power, in the form of specifying an NP-complete problem, such that solving that problem then becomes a way to encode additional transactions into the blockchain (instead of meaningless hashes), then the price of computing power would be the intrinsic worth of pseudo-BTC.