My wife babysits for our grandson 4 days per week while my daughter works.
She babysits at my daughter's home for free, but we are reimbursed $12 per day for fuel to drive 52 miles per day.

My daughter would like to claim the child care tax credit.

Do we then have to claim this as income and pay social security and income tax? I think my wife would be an "independent contractor" and treated as if she were in business for herself. Can we then claim the mileage as an expense and zero out the income for tax purposes? I just don't want it to cost us more money or trigger an audit.

“My daughter would like to claim the child care tax credit.”---> First of all, your daughter doesn’t owe the taxes, Social Security and Medicare taxes (together referred to as FICA tax) and the federal unemployment, or FUTA, tax UNLESS annual pay crosses a set threshold, which was $1,700 for 2010. Your daughter can claim her dependent and child care tax credit on Form 2441 on expenses that she pays her mother to come to her home or taking her child.
“Do we then have to claim this as income and pay social security and income tax?”---->No; as she submits her auto expenses to her daughter and are reimbursed for actual expenses, the reimbursements are not considered income ( actually your spouse didn’t earn it). So, you are not subject FICA taxes and income taxes on the reimbursed expenses. As your spouse provides her daughter with free babysitting, there is no taxable income generated.
“ I think my wife would be an "independent contractor" and treated as if she were in business for herself. Can we then claim the mileage as an expense and zero out the income for tax purposes?”----> It depends; automobile Expenses directly attributable to the conduct of business( I mean between business an business) are deductible; however, commuting expenses (between her home and regular business location) are not deductible. To qualify as an independent contractor, your spouse must show factors that indicate true independence, such as being able to determine when (days or time) and where work is performed, to work for others and provide their own equipment. An example of a household independent contractor is a gardener who comes to the home as needed or on a flexible schedule. In general, nannies are almost always employees of the families they work for. Common law makes the determination of when an employer/employee relationship exists. Under common law, a worker who performs services for you is your employee if you can control the work schedule, what will be done and how it will be done. It does not matter whether you give your nanny great latitude, but rather that you have the right to control the work. It does not matter if the work is performed on a full time or part time basis. It does not matter whether the worker lives with you or not. It does not matter if he/she is paid hourly, daily or a salary. It does not matter how the employee refers to herself or how you refer to him/her in an employment contract. The IRS articulates strict guidelines that differentiate employees and independent contractors to further eliminate confusion (Refer to IRS Publication 926). Nannies, elder care givers, maids, housekeepers and other domestics are generally considered employees. You are generally obligated for all payroll tax filings and remittances if you pay the worker $1700 or more in the calendar year.