I have a qualifying dependent, but no name is listed in the dropdown box in the Dependent Care credit page. Any ideas?

Return to the My Info section of TurboTax to ensure the dependent is correctly showing as your qualifying dependent.  Occasionally an improperly marked question here can cause the dependent to not be qualified. 

Sometimes, it is easiest to remove the qualifying dependent and re-add them into the program to clear any errors.  Just be sure to note the social security number before deleting for ease of re-entry. 

Have you entered any earned income on the tax return?  If you are married, have you entered any earned income for your spouse on your tax return?  Is the child under the age of 13?

Make sure you have entered your W-2's and/or self-employment income before you try to work on the childcare credit.

https://www.irs.gov/help/ita/am-i-eligible-to-claim-the-child-and-dependent-care-credit