BME is a newer smaller field than ME or EE, and different schools have different flavors of BME, bioengineering, biochemical engineering, etc. Some are more and some are less medical in scope or flavor.
I think rankings for this field do differentiate programs that are serious and have research and access to professors that are really interested in BME, but of course that does limit your choice of schools and thus scholarship opportunities, location and other things that the larger ABET accreditation criteria would lead you to.
Also if you are also interested in say aerospace engineering and BME, make sure both programs meet your interests. This means paging through lots of web pages, but hopefully this will be both interesting and informative. A few schools may fall off your list, but maybe googling something you see and like will add a school.
I would even recommend looking at actual class schedules (typically school X biomedical engineering four year plan is a good search term) to see if the coursework is interesting to your student.
Having close proximitiy to a research hospital may indicate research opportunities and also allows easy shadowing and other medical ECs that you might want if your interest wavers from BME to med school.
Having flexible programs may allow you to either double major in say ME and BME or get a degree in ME with lots of BME classes or whatever.
I also wrote this up a few months ago. It also discusses getting a more traditional degree, for job prospects maybe or to leave more options, and weaving an interest in BME into your studies.
The top BME programs now specialize in tracks, ME, EE, CS, or Mat’l Science. If you look at those, you can get some idea on how to weave an interest in BME into your undergraduate degree, and also see what a specific school offers. I think these programs are also producing graduates that people would want at the B.S. level. BME is a fairly new and evolving field, so I think some of the nay-saying is based on industry trends that are years or even decades old.
ChemE programs often specialize in process engineering, aka chemical plants or biological processes, although every school has somewhat different flavor. ChemE programs require a lot of chemistry classes and have some of the best match for pre-med requirements such as organic and biochemistry. However, the actual junior and senior year project type classes may or may not be of interest to you or applicable to your BME graduate studies. Similarly biochemical engineering, bioengineering, biomolecular engineering have very little to do with the human body, but may be very applicable to say pharmaceuticals.
Payscale salaries for BMEs are very competitive with other engineering disciplines at the BS level. And who knows if petroleum engineers will really be able to command high salaries now that oil is cheap again. Similarly aerospace pays well but is not thriving or expanding.
For now, your interest in BME and/or pre-med can help you select a good match college. Some schools have limited enrollment into engineering programs as well, and BME can be even more competitive at say UCSD.