I use the 1:10000 rule (https://xkcd.com/1053/) for this argument. It is my opinion that the intrinsic value of bitcoin is far far greater than anyone really comprehends. People generally are too focused on people just buying and selling the asset, but not about the cryptographic proof, and incentive structure that perpetuates the network functioning. So when people like to ask what the intrinsic value of bitcoin is, I say :

You are thinking of the store of value property without the cryptographic property that secures it. You are talking about it as one would talk about valuable shells, and even shell sellers, and not the security that separates it from all forms of value exchange before it. To understand the intrinsic value of bitcoin it is good to use the analogy of an envelope. So i ask you : does an envelope have intrinsic value?

If I have a letter that I need to send securely, would I prefer to send it open, and ready to be read? Or would I prefer for it to be sent within the confines of an envelope? Now let's think... would I prefer to just leave an open envelope, or would I think it necessary to put some seal across it to ensure that it isn't opened, and read whilst en-route. Does a sealable envelope have more intrinsic value than an unsealable one? Hmm... but what if someone can counterfeit the seal? I need some method of ensuring that the message I want to get to that person isn't read, and preferably, isn't even identified as a message. That's when you need to get lawyers involved. Sounds expensive. There's obviously something intrinsically valuable about such a service, because many people pay oh-so-very-much money for it.

Now let's imagine we're not talking about letters, we're talking about money. Would you address a $100 note through the mail to your child for their birthday? You'd pay for a stamp at least. Would you encase it in an envelope that you paid for at the local post-shop? Hmm... bill in the envelope. Anyone will be able to see that it is money. Perhaps encase it in a card? What if it is $1000? $10,000? $10 million? Does the protection service of an envelope somehow decrease because the value of the contents of the package increases? It's the opposite, isn't it? Bitcoin is both the cryptographically sealed envelope, and the network to deliver it.

What say you? Is an envelope intrinsically valuable?

Strictly speaking "intrinsic value" does not exist, there is only subjective value placed on things based on their properties and usefulness to humans.

Gold only has "intrinsic value" because we value that we can use it to build machines or shiny jewellery. But that is only valuable because we value machines and shiny things. Even air is only valuable to us because we require it to survive, but anaerobic lifeforms don't care. So the value is not intrinsic, but only subjectively "intrinsic" because most people value staying alive.

So the subjective value of Bitcoin is based on its its properties. Humans happen to use money, and the properties of Bitcoin could theoretically make it "better" money than any money that has existed so far.

Anything can be money, but is judged on how "good" a money it is depending on several criteria such as:

1.) Scarcity: Only a select group can create money.

2.) Durability: Should be resistant to aging and natural elements.

3.) Divisibility: Can be divided into smaller sub-units.

4.) Transportability: Easy to transport and transact in.

5.) Recognizability: Should be difficult to counterfeit.

6.) Fungibility: Any $1 bill is equal to any other $1 bill.

Bitcoin could theoretically be better in every category.

However, it is still suffering from lots of problems.

Currently "portability" is often suffering due to network congestion and high fees, which also impacts "divisibility" as it becomes impractical to move small amounts.

However, there's still a good chance that either Bitcoin Core or Bitcoin Cash or any other of the current cryptocurrencies will find a way to scale and retain good portability and divisibility.

tl;dr Bitcoin has no "intrinsic value", nothing does, but the subjective value is that it could be better money than any other.