Can I re-use an email address for a new Apple ID that was previously a secondary email address? I want to use my son's email for his new iPad mini, but previously his email was associated with my Apple ID on my iPhone 5c as an alternate email address. I have deleted his email from my Apple ID, but I still can't set up his iPad using his own email address.

I am attempting to create a new Apple ID for him with his email so all my contacts won't come up on his iPad mini. When we try it, it says that the email is already associated with an Apple ID.
Yes, you should be able to. Having an email address as a secondary email address on another Apple ID does not mean that the email address does not have an Apple ID already associated with it. Try to regain access to the existing Apple ID associated with the email address at iforgot.apple.com.