I’m trying to claim the childcare tax credit, but my son’s name won’t pop up in the drop down list. His dad is claiming him this year, but I’m the one who paid childcare?

If you are not claiming the dependent, you can't take that credit.   The only parent who has the right to claim a child as a dependent is the parent whom the dependent spent the most nights with.  That parent (the custodial parent) is entitled to the Child Tax Credit, and also to file Head of Household, claim the Earned Income Credit, and the Dependent Care Credit.  The other parent (the non-custodial parent) cannot claim the child.  

If the custodial parent is supposed to give (or wants to give) the dependent exemption to the non-custodial parent in certain years, that is done by the custodial parent giving the non-custodial parent a signed form 8332 Dependent Release.   This allows the non-custodial parent to claim the Child Tax Credit and exemption only.  The ability to qualify for Earned Income Credit, Head of Household, and the Dependent Care Credit, always stays with the custodial parent and can't be waived, shared, or transferred.

Where can you find 8332?

You are assuming he is the custodial parent with your answer. We are both custodial parents though, meaning we have both joint legal and physical custody, and I am the only one who pays for childcare. It even stated that I can claim the childcare credit even if I’m not claiming him this year, which is why I asked the question. 

If the child was with each parent for an equal number of nights, generally the custodial parent is the parent with the higher adjusted gross income. The noncustodial parent can't treat the child as a qualifying person (and therefore, cannot claim the Child and Dependent Care Credit), even if that parent is entitled to claim the child as a dependent under the special rules for a child of divorced or separated parents.

For more information, please see Who is a Qualifying Person for the Child and Dependent Care Credit?

@presleyw

Are you married?  You can't get the Child Care Credit if you file Married filing Separate.  You need to be Single or Head of Household.  How old is your son?  Has to be under 13.

We are not married and he is 6.