What were the requirements for being member of the Sanhedrin?
You need to remember that the Sanhedrin was the creation of the Jews. It was justified or based on the seventy elders who aided Moses (Numbers 11:16-17), but where these men individually served as judges to relieve Moses of the burden, the Sanhedrin formed a single court. Mostly likely the Sanhedrin evolved from the elders after the return from captivity (Ezra 5:5, 9; 6:7). The book of Maccabees mentions that a senate was formed composed of the elders of the nation (I Maccabees 12:3, 6; 14:20; II Maccabees 1:10; 4:44; 11:27). It was at this time that the high priest became the presiding officer of the council.
It appears that the early courts were made up mostly of priests who received their position by appointment by members of the older families in Israel. About 76 B.C., this changed so that Pharisees and Scribes could have seats on the court. Still it appears that the ancient families held control over the appointments. Under Herod, the old nobility lost their influence, and appointments began favoring the Pharisees. At the time of Jesus, the court was composed the high priest, members of the families from whom the high priest was selected, elders of the tribes and notable families, and notable legal experts.
Supposedly, when a vacancy appeared on the court, three people could propose a replacement, so long as one of the three was a current member of the court. There were qualifications, but how well they were followed is not known. The qualifications were based on "Moreover you shall select from all the people able men, such as fear God, men of truth, hating covetousness; and place such over them to be rulers of thousands, rulers of hundreds, rulers of fifties, and rulers of tens. And let them judge the people at all times. Then it will be that every great matter they shall bring to you, but every small matter they themselves shall judge. So it will be easier for you, for they will bear the burden with you" (Exodus 18:21-22). And, "Choose wise, understanding, and knowledgeable men from among your tribes, and I will make them heads over you" (Deuteronomy 1:13). The preference for priest was based on, "And you shall come to the priests, the Levites, and to the judge there in those days, and inquire of them; they shall pronounce upon you the sentence of judgment" (Deuteronomy 17:9).