Why am I not getting the Other Dependent Credit for my 22 y/o son who is a full time college student?

Probably because you did not indicate that he lived in your home for the Whole Year.  An absence from the home due to education is temporary so enter the Whole Year 

When you entered his information in My Info, did you say he is a full-time student?   Enter his SSN?   When it asked if he provided over half his own support ----an oddly worded question--- did you say NO?   

Have you entered all of his personal information, including his SSN, and did you indicate he lived with you all year?   What is your AGI? 

To be eligible for this credit, the person(s) being claimed must fit the definition of a qualifying child or a qualifying relative, as defined here:

Qualifying child

Qualifying relative

The credit is $500 per qualifying dependent as long as the adjusted gross income (AGI) doesn’t exceed $200,000 ($400,000 if filing jointly). The credit goes down $50 for every $1,000 that the AGI exceeds the $200,000/$400,000 limit.

$500 Credit for Other Dependents

He must qualify as your dependent. Being a full time student is just one of several rules. 

If you income is too high, you may not qualify (over $200,000, $400,000 married. .  Otherwise, you've answered something wrong in the interview.

There are two types of dependents, "Qualifying Children"(QC) and Other ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, student status, a relationship test and residence test.

A child of a taxpayer can still be a “Qualifying Child” (QC) dependent, regardless of his/her income, if:

1. He is under age 19, or under 24 if a full time student for at least 5 months of the year, or is totally & permanently disabled
2. He did not provide more than 1/2 his own support. Scholarships are excluded from the support calculation
3. He lived with the parent (including temporary absences such as away at school) for more than half the year

So, it doesn't matter how much he earned. What matters is how much he spent on support. Money he put into savings does not count as support he spent on himself.

The support value of the home, provided by the parent, is the fair market rental value of the home plus utilities & other expenses divided by the number of occupants.

The IRS has a worksheet that can be used to help with the support calculation. See: http://apps.irs.gov/app/vita/content/globalmedia/teacher/worksheet_for_determining_support_4012.pdf

See full dependent rules at: https://turbotax.intuit.com/tax-tools/tax-tips/Family/Rules-for-Claiming-a-Dependent-on-Your-Tax-Return/INF12139.html