No, you cannot claim your friends child simply because you are providing child care whether or not you have income. Her mother is the one providing for her financially and she lived with her for over half the year, so her mom gets to claim her.
Also, there is no benefit to claiming someone if you have no income.
However, what you are describing is bartering, and it would be considered taxable income. It is possible that your friend should be claiming you as a household employee and pay the employer share of SS and Medicare taxes. So in reality, you should have taxable income, even if it is not cash, it would still be earned income and you could possibly (if the income is low enough) qualify for the Earned Income Tax Credit.
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