I have been advised to mention faculty members in the area I want to apply for (ECE PhD). My Question is: how much of an effect would this have on my admission? Maybe that particular faculty will not want to work with me but some other person in a somewhat related area will. Also how flexible would my choice of specialization be, when I am admitted to a university?
Although it depends on the particular institution, it can have a positive effect. If you do mention a faculty member as a potential adviser, be sure to briefly state why. Usually your explanation would be that you fancy the idea of doing research in XYZ and she is an expert in that area, or that you've read her papers about XYZ and you find them exciting.
Some research departments look beyond your academic qualifications, and "fit" is one of the possible criteria. Knowing that you've actually taken the time to look at people's research agendas and that you've concluded that there's a couple of faculty members you'd be happy to work with is a good sign.
Be mindful not to elaborate on things you don't have a good grasp on in an attempt to impress, as this can have the opposite effect.