I would seek the advise of a CPA so that you fully understand the state and federal tax implications of such a decision. Additionally, I have some questions regarding where the other parent is (or even if another party has a legal parental relationship with your daughter that would require their consent as well). There are two different questions here. One, can you legally allow someone else to claim an income tax deduction for your biological daughter from a family court perspective, which I can't answer from the facts you provided and two, is that even legal which is why I am directing you to a CPA.
First, do you receive/pay child support for your daughter? If you are divorced from her dad, then that question must be answered, and dad consulted, before a decision is made. If child support is received, you must immediately notify the payor that your daughter is not in your home and modify/revisit the order.
No, the friend's mother cannot claim your daughter on her income taxes, since your daughter is not a legal dependent of the friend's mother. Only a parent can claim her as a legal dependent.
Please note that this answer does not constitute legal advice, and should not be relied on, as each situation is fact specific, and it is not possible to evaluate a legal problem without a comprehensive consultation and review of all the facts and court pleadings filed in the case. This answer does not create an attorney-client relationship.