I hear you iab. I think my allergic reaction was to the "would increase enrollment" part. It would depend on how that was unpacked. Is it "get more tuition" or is it "better align prospective students to the path that most fits them"... If the second, then I think it goes way beyond a name change to education and engagement with primary and secondary school students, community outreach, digital outreach... I have taught from time to time, most recently at New School of Architecture and Design here in San Diego. They have a "Sketch Jam" series that invited junior high and high school students into the studios on Saturday to get exposed to different departments; industrial design, graphic design, interior architecture, architecture. I think this has been a meaningful program for them. They also connect to area community colleges. A few primary schools in the region have also reached out to me to come in and explain design. I think this might be even more meaningful as a relationship with a school that could have more of a sustained effort than my schedule permits.
Lastly, I don't think the idea of changing the name of industrial design is a new idea. SVA has an "objects of design" program, SAIC and Cranbrook have their "3D program", and of course Stanford has the D School. My impression of these programs, and these might be my impressions, is that they are not really focused on producing industrial designers.
So I think it depends on what the goals of the program are.
If the goal is to have a more traditional ID program (define that as what you will in this ever changing world), and to enroll more students (and more dedicated students), then my opinion would be to focus more on pre-enrollment engagement, which is a more difficult and sustained task then changing the name.
If the goal of the program is to focus on less traditional aspects of design (interaction, strategy, design thinking, service design, user experience) then I think both industrial design and product design might be too limiting in their definition.
Just one designer's opinion, which is what I think the OP is asking for.