This process is actually fairly well established.
Step 1 - Placement of a name in nomination
Somewhere it says that the Dean nominates new members. In fact, any SC member can place a name in nomination. There was also a procedure put in place by which a citizen can present a petition to the SC with five citizen signatures on it to place a name in nomination. This was done to provide a way for citizens to bring to the attention of the SC worthy potential members of whom the SC might otherwise be unaware.
Step 2 - SC Approval
Nominees are approved by the SC by simple majority vote of chairs. There is, AFAIK, no written guidance as to how this process happens.
Step 3 - RA Vote of Confidence
Article I, Section 7 specifies that the RA conducts a vote of confidence on SC nominees, based on their "perceived likelihood to uphold the constitution" Precedent to this point (we'll have to dig through transcripts) includes:
1. The RA may invite, but may not compel the nominee to appear and/or answer questions
2. The vote is simple majority
3. Were the vote of confidence to fail, there is, I believe, a tacit understanding that the nominee would not then become a member of the SC, though this has never actually happened, to my knowledge.