Last year my wife and I had three sources of income: 1) an IRA distribution, 2) Unemployment Compensation, and 3) Social Security. We would like to make a regular IRA contribution. Do we have "qualifying income"?

No - you cannot contribute more than your "taxable compensation" - which includes such things as wages, tips, commissions, and self employment income. It appears you had no taxable compensation in 2011, so you cannot make a contribution to your IRA.