Way to start planning early!

A friend of mine just took the LSAT recently and I have been giving her all sorts of advice on this throughout her prep process, so this is going to be a long reply because I sympathize and always want ambitious test preppers to succeed!
The TL;DR version: Economist and New Yorker are fine, but I recommend supplementing with dry, scientific scholarly journals (The Annals of [some medical field], for example) and periodicals, like Scientific American, because most people struggle with the science passages above all else. Work on concentration endurance by reading long articles. If you anticipate that reading comp will be your worst section, make sure you do every practice test out there, focusing on comparative reading problems. The Power Score RC Bible may help you too.
The long version:
First, make sure you have time to drill every practice test out there, as above posters have indicated. Second, get the PS RC bible (along with every other bible--they're amazing!). Third, Economist and New Yorker are decent choices but, since you're going this route, you should add Scientific American, other science periodicals, and, if you have access to scholarly journals through something like JSTOR (or even Google Scholar). Read the technical/medical/hard science articles and then try to summarize them to test your own comprehension. This practice will also help you down the line when you're in law school and learning to brief cases as a green 1L.
The problem with RC today is that you must remember to train for the comparative passage reading too, and you can't really do that unless you drill actual practice tests--there's no substitute for that unless you want to go through the end notes of a scholarly journal article, compare two cited works and analyze their approaches/theses/argument structures. That would be crazy and unreasonable overkill though, unless you're still in college/grad school and you actually need to do this for a term paper or your own senior thesis or something. If that's the case, then you can efficiently accomplish your research and prep goals simultaneously.
The only real benefit to reading dry material is to get a feel for the language and to work on your ability to focus on, and comprehend, esoteric scientific/literary concepts immediately and without re-reading, under all sorts of unfavorable conditions. Most people have the hardest time comprehending the science-related passages, because even wacky political arguments or obnoxious literary reviews are at least tangentially approachable. Science may stump/scare people because of limited exposure to the fundamental concepts in the passage, despite the fact that every passage is written in a way that should be easily comprehended by the average reader of a NYT article.
The other issue with RC, of course, is that your level of focus will dwindle as you proceed with the exam, so one benefit of reading very long articles is to work on your concentration endurance and make sure that you know what's going on 20 pages into an article. This is also total overkill, but if all you are doing at the beginning of your prep work is prepping by reading periodicals and scholarly stuff, then this is probably the best (though inefficient) way to simulate fatigued reading conditions which will be present on test day.
Sometimes the passage will be straightforward despite its astrophysics/molecular biology topic, but it will be hard to chew through anyway if it's the 3rd passage in the 4th section of the exam. In that sense, you're kind of playing the RC lottery. Personally, I was generally okay with RC score-wise, and I was not intimidated by the science topics, but I hated it more than any other section due to sheer boredom (I really enjoyed the LR and LG because they were fun, while RC felt like an SAT reboot). On my first LSAT attempt a few years ago, I "lost" the RC lottery and got stuck with RC in the second half of the test, at which point I was fighting through a blinding migraine and could not concentrate at all...Although I ended up canceling my score, re-taking, doing well, and happily enrolling at the school of my choice, I'm still shuddering over some horrific passage about the magnetic forces beneath sea beds from several years ago just because I knew that, under better conditions, I would have done fine the first time.
So, to be prepared for the worst case scenario of being tired, possibly blind in one eye, not being able to focus, and having to navigate simple arguments comprised of polysyllabic words on foreign subject matter, you might want to delve into the Annals of Molecular Biology early on, and then proceed with the standard TLS-prescribed routine of drills, PTs, bibles, and the like. Best of luck to you in your prep!