I'm not an admissions officer by any means, but I've done a good bit of research on the admin process, and having just gone through the admissions cycle myself, your major isn't a deciding factor when it comes to getting into law school. All things being equal, if you have two candidates, one with a 3.8 Poli Sci GPA and the other with a 3.0 Engineering GPA, the 3.8 will still look better to the vast majority of adcoms. The importance of your major comes into play when the numbers are close, a 3.6 Econ GPA versus a 3.7 Poli Sci GPA, for instance.
The reason for this is the rankings. Higher median LSAT's and GPA's of a school's incoming class increase its ranking score. While itis much easier to get a 4.0 in typical pre-law majors than get a 3.3 in a legit engineering program, majors don't factor into school rankings--and let metell you, they care about those rankings a lot. Rankings are also why something called "yield protection" exists--i.e. rejecting or wait-listingoverqualified candidates because of an assumption that they won't attend. % yield factors into rankings as well...
But truthfully, engineers and scientists complaining about the GPA thing should get over themselves. Yes you had harder classes and studied more in undergrad(I took upper division Berkeley Math so I know a little bit about undergrad hell classes), but knowing how to code triple integral algorithms won't helpyou in Constitutional Law your first year. The Poli Sci or History majors who have already seen the cases you study in Con Law multiple times probably evenhave an advantage over you. The technical undergrad degrees don't really come in handy or become relevant until you start working, a time at which theHistory major who got a 4.0 will have gotten into and graduated from Harvard and will be making more money than you. If you want to be rewarded for yourtechnical major right now, do something that applies it right now.
To speak from experience and people I know, if you want Harvard, Yale, or Stanford and are willing to put all your eggs in one basket, go take an easy major aslong as it has reading and writing in it, get your 4.0+, and devote the time the technical major folks spend in labs to studying for the LSAT and doingextraordinary extracurricular activities and jobs. But you better not bomb the LSAT. If you do, no law school and probably no job because of your easy major.