One of the on'yomi readings of 相 is しょう. If I'm doing my legwork right, when it's read this way, it has an indication of government involvement, in words like (and here, I'm relying on my dictionaries—I have no idea if these words are commonly used?)
丞相 じょうしょう "emperor's assistant"
首相 しゅしょう "prime minister" ← I thought the word for this was 総理大臣 そうりだいじん？
Apart from my prime minister question above, I'm struggling to understand what is really meant by "government minister". Is this a government minister of old? (China, even?) Is this contemporarily used for ministers in the government? What level of minister are we talking about? Any old minister, or is it for special ones?