Iowa

Well, first of all it depends on your interest. Mechanical Engineering is a broad field which includes various domains like design, manufacturing, aerospace, vibrations etc. Avionics is one of the specialized field of Mechanical engineering itself. For pursuing avionics, basic knowledge of mechanical is required.
From the job perspective, mechanical is better than avionics because you are not restricting your domain. Even an industry which requires avionics expert would hire mechanical engineer if he has sufficient skills for that job. But if you take avionics, you are restricting your job opportunities in other mechanical domain jobs. So according to me from job point of view, go for mechanical over avionics.