For me, definitely mechanical engineering.
I trained as a physicist, have my masters in biomedical engineering (sort of, my degrees are complicated) and PhD in a related field. My department actually actively seeks out graduates of more 'pure' disciplines for bioengineering programmes. That is, students of electrical, mechanical engineering, materials science or physics. The reason for this is that mechanical engineers know a great deal about mechanics. Electrical engineers similarly with their field. They can then, relatively easily, pick up necessary biology and apply their expertise thinking like a mechanical engineer.
Bioengineering is, obviously, an interdisciplinary field. I feel that it's more useful to have a pure specialist come in and think the way that they have been trained to. For me, a bioengineering undergraduate is too broad - you're trying to mix in electronics, mechanics, materials science, anatomy, biology and pharmacology all into one pot.
You should have a look at mechanical engineering programmes in detail too - there may be some biomechanics electives that could give you a leg-up when it comes to post-grad.
Addition: In my experience, those who have trained in bioengeering at undergraduate (at good universities, might I add) are missing key things like ability with CAD software, FE analysis and general programming/modelling.
Lastly: obviously I have my preference. This, clearly, doesn't make for a rule. If you look at the bioengineering course and decide it looks amazing - then it wouldn't be unreasonable for you to go for it - you will still have plenty of options afterwards. For instance, (you don't say where you are from) but if you're from the UK then the NHS bioengineering programme actually requires an accredited undergraduate (or post-graduate masters) degree.