I am in favor of replacing the tag bitcoins or bitcoin with the more appropriate and specific tags proof-of-work and cryptography, but I am currently less in favor of using the tag cryptocurrencies.
The word 'bitcoin' refers to a specific (though decentralized) product that you can buy on an exchange. I cannot think of any appropriate tags on this site that refer to specific products, and I do not believe that it is appropriate to have tags on this site about specific products. Tags on this site are uncapitalized, and since the tags on this site are uncapitalized, this site favors tags which are not proper nouns. Bitcoin is a proper noun when referring to the cryptocurrency network, so we should not use the tag bitcoin.
The tag bitcoins does not seem like the best tag since it seems like even the tag bitcoin is an improvement over bitcoins. The term 'Bitcoin' with an upper-case 'B' refers to the entire cryptocurrency blockchain while the term 'bitcoin' with a lower-case 'b' refers to a specific unit of currency worth $28,175.32 as of 5/1/2023 in the Bitcoin blockchain. The term 'bitcoins' is the plural form of 'bitcoin'. We can therefore conclude that the tag bitcoins refers to the plural of 'bitcoin' which refers to a quantity of digital currency instead of the blockchain. It is more appropriate for the tag to refer to the entire blockchain than to a specific unit of currency on that blockchain.
The tag bitcoins is outdated in favor of the more appropriate tags cryptocurrencies, proof-of-work, and cryptography. The tag 'bitcoins' was created in September 5, 2012. In 2012, Bitcoin's market capitalization completely dominated the total cryptocurrency market capitalization. Only in 2017 did we see Bitcoin's percentage of the total cryptocurrency market capitalization drop from about 85 percent to about 33 percent. Bitcoin's percentage of the total cryptocurrency market capitalization has been around 50 percent since the middle of 2017 (take this value with a grain of salt since it does not take into consideration lost bitcoins). Right now there are thousands of alternate cryptocurrencies to choose from besides Bitcoin, but in 2012, Bitcoin was the only cryptocurrency that was not just an experiment. It made more sense to use a 'Bitcoin' tag in 2012, but now it does not make sense to use the bitcoin tag in favor of cryptocurrencies.
The site https://crypto.stackexchange.com/tags does not have a tag for Bitcoin. The Cryptography Stack Exchange site has thought about the usage of the Bitcoin tag, and they have decided that it is best not to use a 'bitcoin' or 'bitcoins' tag, and I agree with their determination.
There are several StackExchange sites dedicated to cryptocurrency technologies including Bitcoin, Cardano, Cryptography, Ethereum, Iota, Monero, Stellar, and Tezos. Since the rest of the StackExchange network embraces cryptocurrency technologies, MathOverflow should at least be welcoming to those who ask high quality mathematical questions related to cryptocurrency technologies on this site. Those other sites have not attracted low quality posts, so I doubt that a cryptocurrency tag would attract much more than its fair share of low-quality posts. But I understand that cryptocurrency technologies have been controversial and have unfortunately attracted too many bad actors, so some people on this site may not want anything to do with cryptocurrency technologies. I am in favor of using the tags proof-of-work or cryptography instead of cryptocurrencies since the cryptocurrency related questions on this site have all been about proof-of-work problems which are cryptographic protocols; it is better to use a more specific tag rather than the tag cryptocurrencies which the people on this site have different opinions about. There may come a time in the future when we will need a cryptocurrencies tag for mathematical research questions about cryptocurrencies that are not about proof-of-work or any other kind of cryptographic protocol, and we should welcome such questions (even those who are not in favor of cryptocurrencies should at least be in favor of improving cryptocurrency technologies or at least solving a mathematical problem).
Proof-of-work problems are examples of cryptographic protocols, so we may also just use the tag cryptography instead of proof-of-work. There is currently no proof-of-work tag, so we should ask ourselves whether we want to create the new tag proof-of-work or use the existing tag cryptography.