The title of your post suggests that maybe you don't understand the issues here. In the US, for tax purposes, it is better to have a higher cost basis. You don't want to reduce your cost basis. You want it to be as high as possible, within the parameters of the law.
When Coinbase says he will have to pay tax on the whole thing because the cost basis cannot be established, what they mean is that they will have to treat the cost basis as zero in their accounting system. If he then sells or spends the bitcoin, at the end of the year, they have to issue a US tax form showing that he sold that amount of bitcoin, and they will report the cost basis as zero on the form that they send to him and to the IRS. Their accountants have told them that is how it has to be reported.
This occasionally happens with stock. Before everything went electronic, it was actually fairly common that someone would inherit stock from a deceased family member, and they would get paper stock certificates that were stored in a safe somewhere. They would then deposit those certificates with a broker to sell them. The broker would have no way of knowing what the cost basis was.
If your friend has his own records to prove what his cost was for the bitcoin, then he can report that cost himself when he files his tax return for the year in which he sells or uses the bitcoin. He should not have any issues in the year that he performs the transfer to a US account. The issues will come up in the year that he sells or spends it. But if he has records to determine his cost basis, he can use that basis on his tax return. He should hire a qualified tax pro who understands US tax regs for crypto.
If he mined the bitcoin, that does make it difficult to determine the basis. Whatever number he uses for basis on his tax return in the year that he sells or uses it, he needs to be able to back it up with solid documents, because the IRS loves to audit bitcoin transactions.
I am a tax advisor in the US.
BMK