My 16 year old is going to make more than 5k from her part time job. Can I still claim her as a dependent?

[DoninGA]

Since she is under the age of 19 her income is not relevant when it comes to claiming her as a dependent.  You should be able to claim her under the Qualifying Child rules if she meets all the requirements.

To be a Qualifying Child -

1. The child must be your son, daughter, stepchild, foster child, brother, sister, half brother, half sister, stepbrother, stepsister, or a descendant of any of them.
2. The child must be (a) under age 19 at the end of the year, (b) under age 24 at the end of the year and a full-time student or (c) any age and permanently and totally disabled.
3. The child must have lived with you for more than half of the year. Temporary absences while away at college are considered living with you.
4. The child must not have provided more than half of his or her own support for the year.
5. If the child meets the rules to be a qualifying child of more than one person, you must be the person entitled to claim the child as a qualifying child.
6. The child must be a U.S. citizen or U.S., Canada or Mexico resident for some portion of the year.
7. The child must be younger than you unless disabled.

[VolvoGirl]

For her income next year she might have to file her own return.......

They don't have to file if they only have W2 income under 14,600 but can file to get back any withholding taken out in boxes 2 or 17. But you don't get boxes 4 or 6 back. If they got a 1099Misc or 1099NEC they have to file it as self employment income no matter how small the amount.

Be sure on their return they check the box that says they can be claimed on someone else’s return.

[Hal_Al]

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 and/or student status, a relationship test and residence test.