My son claimed himself and just over $5000 in college tuition expenses . Would it be a bigger tax return for me to claim him and his college tuition on my own return?

It depends.  Your son's taxable income, your adjusted gross income, and whether he is eligible to be your dependent as a qualified child enter into the equation.

The nonrefundable and refundable portions of the AOTC your son would get may be more than what you get for the AOTC and the Additional Child Tax Credit. 

Because your son has already filed a tax return, the most practical thing to do, before you file your return, is to prepare the returns "both ways" and see which filing option is best for all concerned. 

You son does this comparison by starting the amendment process, and making the changes to take out the education expenses and change his personal info to note that he is claimed as a dependent on someone else's return.  Note the difference between the original and amended amounts.

You do this comparison by first preparing your return without his education expenses or him as a dependent, and then adding him as a dependent and including his education expenses.  The differences between the two sets of returns for you and your son will answer your question.  Then, be sure to delete the education expenses from the tax return(s) that won't be claiming education expenses.

Note also that if your son provides more than 50% of his support (including education expenses) you can't claim them as a dependent.

Table 2-1. Overview of the American Opportunity Credit for 2019" from IRS Publication 970 at this link

Table 5 from IRS Publication 501 is a good overview of what it takes to claim a dependent.