Does mentoring a summer student during a postdoc help get tenure-track position?
If this is all you've got that go ahead and put it in your applications.  However, you'll still look very inexperienced in teaching in comparison with applicants who've actually taught classes.

The question is about teaching experience and positions in the US at institutions that are somewhere between liberal arts colleges and R1 universities.  The question doesn't mention the discipline, and this could be important.  

I've chaired several tenure track search committees in mathematics over the last few years at such an institution (my institution is a "School of Mines".)  We get hundreds of applications for every tenure track position and everyone we've hired recently has had significant teaching experience, typically including working as a TA during graduate school and then doing additional teaching of classes as part of postdoc or visiting assistant professor positions for one or more years.  

In my experience, new tenure-track hires in other science and engineering disciplines tend to have somewhat less teaching experience than in mathematics.  However, I'd say that most new hires at my institution have at least one or more semesters of experience teaching classes.