why my son, a student (I'm paying more than half of living expenses plus tuition) but also working ( earnings over $5K) , is considered a non-dependent by turbotax?

You haven't provided enough info to answer that question.  If he does qualify, you most likely answered something wrong in the interview.  One common error: if he is away at school, you still answer that he lived with you all year. 

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.

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

How old is your son?  If he is over 23, then he would need to earn less than $5,050 in order for you to claim him as a dependent even if he is a full time student. 

If he definitely meets the criteria below to be considered your dependent, go back through and double check your answers.  Be sure that you mark the check box stating he is a full time student IF he is 23 or younger. 

The following criteria must be met to claim someone as a qualifying child:

• Your child (including adopted and foster children), your sibling, or a descendent of any of them.
• Age 18 or younger at the end of the tax year OR under 24 (and younger than you and your spouse) if they are a full-time student
• Lived with you for more than 6 months during the tax year
• They did not provide more than half of their own support (social security does not count)
• They did not file a joint return, unless it was to claim a refund

To claim someone as a Qualifying Relative, they must be:

• Your child ( including step children, adoptive children and foster children) or a descendent of them
• Your sibling (including half siblings) or a child of your sibling or a sibling-in-law
• Your parent or grandparents, including step parents and in laws
• Any other person that lived with you for the entire tax year
• Not a qualifying child of another taxpayer
• Someone that you provided over half of their support for during the tax year