The IRS has made clear that the use to which the profits are devoted does not determine the applicability of the unrelated business income tax.  The tax is determined on the basis of the business activity that creates the income, not the use of the funds.

If I understand your situation correctly, you probably have some unrelated business income that is taxable and some that is not.  “Mining” bitcoin is an active process in which you would actually create a thing of value to be sold to others.  If you did it for a year, that seems to be an active trade or business that is regularly carried on and unrelated to your exempt activity.  Therefore, it seems likely to be subject to UBIT on the profit.  The costs incurred in mining the coins are deductible from the gross income, of course, and will reduce the net income on which to pay the tax.  You can still use the profit for your charitable purposes, just not so much of it.

Investing and trading in other cryptocurrencies, however, is probably like investing in securities, which charities do all the time with their endowments and reserve funds.  Income from investments is one of the passive “modifications” which are excluded from unrelated activity and are not subject to UBIT.  (See Ready Reference Page: “Nonprofits Often Worry About UBIT”)  Gains and losses from the sale, exchange or other disposition of property (other than property that would be properly in “inventory” or held primarily for sale to customers in the ordinary course of a trade or business) are specifically excluded from unrelated business taxable income and not subject to UBIT.

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