Do I need to claim anything on taxes for a family member who moved in and is contributing to household expense?

If she is part of your household, there is no tax consequence to her giving you money toward household expenses, and this does not appear on your tax return.

If she is not part of your household, you are renting. If you are renting below market rates, the money she provides you is miscellaneous income. You can deduct expenses related to this income on Schedule A up to the amount of expense.

If you are renting at market rates, use Schedule E.

Rental income is incurred when a person “contributes a small amount per month” to occupy property you own. Deduction of expenses is determined by multiple factors. I recommend that you consult with a tax expert possessing Enrolled Agent certification to properly identify your eligible tax deductions. Tax professionals searching this forum may want to learn more about complex elements related to rental income by exploring further education at http://fastforwardacademy.com/enrolled-agent-exam-...

You write that the family member is living in an “inlaw apartment.” This implies that it is a separate dwelling unit. However, you state that it is “attached” to your home. A key feature of this matter is whether your family member is occupying a separate unit or the entire house with you. If you share your home, the normal process is dividing your total expenses between the rental use and the personal use based on the number of days used for each purpose. Renting to a family member for less than fair rental value counts as a personal use day. Again, hiring an Enrolled Agent assures that you obtain accurate tax calculations.

if you rent to a member of the family at less than the market value of the rental, you claim the income but you are not allowed to claim any expenses

she obviously is not paying any rent

No, she's sharing expenses, not paying rent.