Can I still claim my daughter if she worked part time and had a child

It depends. See full rules below. 

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 considered third party support and not as support provided by the student.

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 him self.

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

Furthermore, there is a rule that says IF somebody else CAN claim him as a dependent, he is not allowed to claim his own exemption. If he has sufficient income (usually more than $6300), he can & should still file taxes; he just doesn’t get his own $4050 exemption (deduction). In TurboTax, he indicates that somebody else can claim him as a dependent, at the personal information section.  

Even if he had less, he is allowed to file if he needs to get back income tax withholding. He cannot get back social security or Medicare tax withholding.

There are two types of dependents, "Qualifying Children"(QC) and standard ("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 a relationship test.

A person can still be a Qualifying relative dependent, if not a Qualifying Child, if he meets the 6 tests for claiming a dependent:

1. Closely Related OR live with the taxpayer ALL year

2. His/her gross taxable income for the year must be less than $4,050 (2016)

3. The taxpayer must have provided more than 1/2 his support

In either case:

4. He must be a US citizen or resident of the US, Canada or Mexico

5. He must not file a joint return with his spouse

6. He must not be the qualifying child of another taxpayer