Yes you can on a Form 1040 Schedule C subject to certain limitation, but the idea of claiming the mining expenses is to offset the mining income and hopefully have a profit. Hint: utilize §179 deduction to expense all of your initial hardware purchases in the year of purchase. If you are doing something for fun and not worried about losing money, you can't offset these "hobby losses" against your other income. Three years of straight losses and the IRS will assume that it is a hobby. However, the IRS will make you pay tax if you make money on the hobby.
It was a Form 1099 that your wife got. In her case if she made $399 it would still be taxable; the company just doesn't need to report the income to the IRS. Similarly, there is no such de minimis exception for mining income: you make $1 and the law requires your to pay income tax and self employment tax.