Here are my impressions:
Brown has a very flexible curriculum that will basically allow you to study whatever you want and take premed classes alongside that without overstressing your schedule. That can be very helpful if you want to explore different things while still being a premed. They are also notorious for grade inflation (though I hear their organic chemistry class is killer) which can be helpful for medical school applications. They are also a feeder school to a lot of top medical schools, as well as (ostensibly) to their own medical school. I think Brown is likely your best choice.
Duke has a less flexible curriculum and average grade medians (for top schools), so your GPA will likely be lower than it would be at Brown with a similar amount of effort, but not enough that I would use that criteria to make the decision alone. Duke has a strong economics program (obviously their biology program is strong too), high medical school placement (similar to Brown - 80-90%), and is a feeder school to top medical schools. When I went to Blue Devil Days years ago, they told us that Duke applicants above a certain threshold will get automatic interviews at Duke Med. Also if you care about sports, this would be the place for you. I was personally 99% set on going to Duke for undergrad before I changed my mind at the last moment. I think Duke is a very very very close second to Brown for medical school admissions purposes.
Hopkins is a great school, but is grade deflated, screens applicants to medical school (I cannot verify this so maybe current Hopkins undergrads can confirm or deny), and doesn't feed to top medical schools as heavily as Brown or Duke (except to perhaps its own). It has the best public health program in the country, but that shouldn't be the factor that makes your decision. Ultimately, you are going to have to work much harder at Hopkins for the same result than you would at Brown or Duke, and you are more likely to fail to achieve your goal from here. I was accepted to Hopkins, but quickly ruled it out in favor of Duke and the school I ended up attending, and I don't regret my decision. It is a hard school, no doubt about it, but certain types of people thrive there.
Rice is a great school, but doesn't have its own medical school, which is a minus since most schools feed to their own medical school, and doesn't have the nebulous prestige bump that might be associated in varying amounts with the other three. I don't know as much about it as others, but from what I have gathered, it is not as big feeder to top medical schools (though you certainly won't be limited by the school!). I think it should be at the bottom of your list.
I would personally put my list at Brown>Duke>>>>>>>>>Hopkins>>Rice (I think the choice here is really more or less a toss up between Brown and Duke). Ultimately, you should attend the admitted students weekend for all 4 (or perhaps your top 2-3) and see which one you fit into best. That will ultimately play the biggest role in how successful you are rather than the resources of these (utterly fantastic) schools.
Congratulations on having a great decision to make!