Back when I was at an APA approved internship site, our selection decisions were based on the following, in general order of importance:
1. Worked with a mentor who previously sent us successful students, with a good recommendation from this mentor. This usually went beyond a letter of rec- a good mentor would make sure we knew of this student well before application time, such as by introducing us at conferences. Also- good mentors generally don't let unqualified/unprepared students get to the application phase.
2. Applicant's training and career goals in line with what we could provide (if we couldn't get you where you wanted to go, it wouldn't be that nice for us to accept you, would it?)
3. Relevant clinical work with a relevant population (to the major rotation) with a likely ability to function in a minor rotation, with good references from said clinical work. The minor rotation thing could be a big deal at times, particularly when ranking strong candidates.
4. Research and presentation in relevant clinical area
-at this point, we typically had more than enough applicants to fill our slots-
5. Relevant clinical work with a different population
7. Grades...maybe? For most applicants, this was taken care of by #1-4 above. Basically, if you got through 4 years in a good graduate lab, with good refs and some pubs or solid presentations, we didn't really look at grades, assuming that stuff was between you and your mentor. If it got down to us looking at your GPA, you were probably a questionable, low ranked applicant. I don't remember GPA ever being a big factor.
8. Crap totally unrelated to your ability to succeed at the internship, but some faculty couldn't ignore or counteract with logical, informed decision making. These are things like physical appearance, clothing worn to interview, nervousness at interview, etc. We had a decent, well educated faculty who understood the impacts of such factors on decision making, as well as the need for an active process to remove such biases, but they would occasionally sneak through.