Many students like to double major or at least have a minor or two with their major. The student should take a look at what potential minors they would be interested in. It is sometimes more difficult to combine majors from different ‘schools’ or ‘colleges’ within the university.
For example, my S is double majoring in Economics and CS. The College of Liberal Arts at his U lists both those majors however, the College of Engineering does not. For CS, whether he takes it thru COE or CLA, the core requirements are the same. The difference between the colleges are the remainder of required course to obtain a degree from that college. In the case of CLA, typically language is required and from CSE it would be physics. There are also a few differences with the math requirements.
Also note that the College of Liberal Arts is often the largest college - hence, it is typically easier to gain admission to this college over the Engineering College. Also, in some universities, CS is in such high demand, that there is difficulty in obtaining the classes. So, if you are not a designated CS student, you could have lower priority gaining access to classes in some universities.