There is no statutory authority for the Minister of Justice to appoint a judge to a superior court
You edited to clarify that you are "interested in the legislation that allows ministers to appoint the judges." You will not find any. As this answer has always said at its outset, the appointment of judges is made by the Governor General. Neither the Minister nor Cabinet appoints judges.
The rest of the answer addresses what the question originally asked, including the legislation that grants the authority to the Minister and the Office of the Commissioner for Federal Judicial Affairs to organize committees to inform the advice that Cabinet ultimately provides to the Governor General.
Judicial appointments are made by the Governor General
Section 96 of the Constitution Act, 1867 says:
The Governor General shall appoint the Judges of the Superior, District, and County Courts in each Province, except those of the Courts of Probate in Nova Scotia and New Brunswick.
There are similar clauses in the statutes that establish statutory appellate and federal courts. E.g. from the Federal Courts Act, s. 5.2:
The judges of the Federal Court of Appeal and the Federal Court are to be appointed by the Governor in Council by letters patent under the Great Seal.
The Governor General takes advice from Cabinet, a practice that reflects constitutional convention
The Governor General makes these appointments on the advice of Cabinet which in turn is based heavily on recommendations from the Minister of Justice. This practice is justified by constitutional convention1 of responsible government, explained by Andrew Heard:
The legal authority of the Governor General is exercised in practice on the binding advice of a cabinet who can command a majority of seats in the House of Commons. This is the essence of responsible government.
See also, Democracy Watch v. Canada (Attorney General), 2023 FC 31 at para. 9 (a judgment in which the Federal Court rejected a constitutional challenge to the "validity of the Government of Canada’s federal judicial appointments system and judicial elevations system"):
By constitutional convention, when appointing judges to provincial superior courts, the Governor General acts on the advice of the Committee of the Privy Council of Canada. Similarly, the GIC, which appoints judges to the Federal Court, the Federal Court of Appeal, and the Tax Court of Canada, is defined in the Interpretation Act, RSC 1985, c I-21, as the Governor General acting on the advice or consent of the Privy Council for Canada. The Privy Council is composed of all the ministers of the Crown, who meet in the body known as Cabinet (see League for Human Rights of B’Nai Brith Canada v Attorney General (Canada), 2010 FCA 307 [B’Nai Brith] at para 77).2 As such, all federal judicial appointments are made by the Governor General on the advice of Cabinet. In turn, Cabinet acts on the advice of the Minister of Justice [Minister]. (In the case of appointment of Chief Justices and Associate Chief Justices, it is the Prime Minister who provides the advice to Cabinet. For simplicity, these Reasons will refer to the advice to Cabinet being provided by the Minister.)
And see League for Human Rights of B’Nai Brith Canada v Attorney General (Canada), 2010 FCA 307 at para. 78:
In practical terms, then, a statute that vests decision making in the Governor in Council implicates the decision making of Cabinet, a body of diverse policy perspectives representing all constituencies within government.
The Minister of Justice relies on the work of the Judicial Advisory Committees
The powers, duties, and functions of the Minister of Justice are found at s. 4 of the Department of Justice Act and these include having the "the superintendence of all matters connected with the administration of justice in Canada, not within the jurisdiction of the governments of the provinces."
It is through this power that the Minister of Justice is properly involved in the recommendation of people to provincial superior courts, statutory courts of appeal, and statutory federal courts. The Minister relies on the work of the Commissioner of Federal Judicial Affairs and its Judicial Advisory Networks. The Judges Act, s. 74(d) gives the Commissioner the power to "do such other things as the Minister may require in connection with any matter or matters falling, by law, within the Minister’s responsibilities for the proper functioning of the judicial system in Canada."
The history is discussed in A. Anne McLellan, "Foreword" (2000) 38:3 Alberta L. Rev. 603.
Since Confederation until about 1970, the Minister of Justice "generally relied on his personal knowledge or that of parliamentary colleagues of the local Bench and Bar" (McLellan, p. 604).
In the mid-70s, "the Minister of Justice created the position of Judicial Affairs Advisor to provide additional assistance in candidate recruitment" (ibid.).
And in 1988, "a new judicial appointments process was established," whereby candidates would make themselves known to the Minister through the Commission for Federal Judicial Affairs and the Judicial Advisory Committee (ibid.).
1. The critique that a constitutional convention could be contrary to "the letter, spirit or intent of the constitution" is paradoxical, given that constitutional conventions are part of the constitution. As the Supreme Court of Canada said, in Reference re Secession of Quebec,  2 S.C.R. 217: "the Constitution of Canada includes... the global system of rules and principles which govern the exercise of constitutional authority in the whole and in every part of the Canadian state." And that these supporting principles and rules "include constitutional conventions and the workings of Parliament." And in Reference re: Resolution to amend the Constitution,  1 S.C.R. 753, p. 883: "conventions... form an integral part of the constitution and of the constitutional system. They come within the meaning of the word 'Constitution' in the preamble of the British North America Act, 1867... constitutional conventions plus constitutional law equal the total constitution of the country."
2. While in some contexts, the distinction between the Privy Council and Cabinet might be meaningful, in many, including in this context controlled by constitutional convention, it is Cabinet that in fact is left to express the judgment of the Privy Council. The Supreme Court of Canada has written: "In the usual course of things, the Governor General exercises these powers for the Queen in right of Canada, acting on the advice of a Committee of the Privy Council (which consists of the Prime Minister and Cabinet of the government of the day)" (Ross River Dena Council Band v. Canada, 2002 SCC 54).