Blockchains are valuable for a number of reasons, what really makes them special is the decentralized aspect of them. A decentralized and widely distributed blockchain is a very powerful tool for democracy, freedom of expression and information.

Blockchains are valuable because we assign them value just like anything else. They maintain their value because they are cryptographically secure and immutable. You cannot simple print more Bitcoin and must acquire it from someone else or mine it (Proof of Work) some other blockchains require Proof of Stake such as Stellar Lumens and other require proof of care or some hybrid. The worst blockchains have been premined and are for sale to the highest bidder. These favor the wealthy and the creators. They may have assigned economic value but they have little value to society as a whole as the aren't really decentralized and are consolidated in the hands of the few.
Many blockchains are clones, these should be avoided too because they will likely never be as valuable as the original. For example Litecoin will never be worth more than Bitcoin because it is a copy and isn't revolutionary. Ethereum is revolutionary because it is a smart contract, not just a representation of value like Bitcoin. EOS is similar to Ethereum in that it is also a smart contract but will probably never be worth as much because of first mover advantage Ethereum has. The childchains give Ethereum value and the decentralized applications which are powered by Ethereum because they create demand for it, improve it and make it more functional. EOS may never catch up to this as it is a couple years late as not as widely spread. NEO might become more valuable because of political reasons. I can go on and on about various coins, if you want a valuable one, go with XLM, mainly because it's extremely fast, as cheap as possible, backed by industry titans and it's being widely distributed for free. Ultimately adoption and acceptance will make a blockchain valuable.

As for the power of blockchains; having the ability to share and store information which cannot be edited by a third party is extremely valuable. We can use this to digitalize democracy, make accounting and taxes more transparent, make society require less trust and faith in big government, legal systems, banks, and other holders of information. This is revolutionary and blockchain could change society as much as written language. It's a form of communication that cannot be destroyed or altered.