Who could become a medieval minstrel?

Certain people in that time era. Answer: In the middle ages folks usually took up a trade by apprenticeship. That is a professional would take on (buy, be sold, kidnap) a likely youth as a servant. The servant would not get paid, but would be taught the ins and outs of the trade. In the case of the minstrel the apprentice would learn the songs, stories, and how to play an instrument. In addition the apprentice would learn the route, the friendly places to stay/play and become recognized as the senior minstrel's associate. This would allow him to take over when the master grew old.