What is the origin and purpose of a new bishop being questioned by a layperson? At the enthronement of Justin Welby as Archbishop of Canterbury on 21 March 2013, the following exchange took place as part of the ceremony. Evangeline Kanagasooriam is a  layperson from the diocese, and a pupil at a school with a very long association with the cathedral. (The text is from the order of service, p14, and it can be enjoyed on video as well.)

When the Archbishop arrives at the West Door he strikes it three times with his pastoral staff. The doors are opened and a fanfare is sounded.
Evangeline Kanagasooriam: We greet you in the name of Christ. Who are you and why do you request entry?
The Archbishop: I am Justin, a servant of Jesus Christ, and I come as one seeking the grace of God, to travel with you in his service together.
Evangeline: Why have you been sent to us?
The Archbishop: I am sent as Archbishop to serve you, to proclaim the love of Christ and with you to worship and love him with heart and soul, mind and strength.
Evangeline: How do you come among us and with what confidence?
The Archbishop: I come knowing nothing except Jesus Christ and him crucified, and in weakness and fear and in much trembling.
Evangeline: Let us then humble ourselves before God and together seek his mercy and strength.

I believe that this is new for the occasion. The order of service for Rowan Williams in 2003 does not contain it, but it does have the triple knocking followed by a welcome by the Dean (which also occurred slightly later in the Welby ceremony).
Is there a precedent for this question-and-answer text in Anglican or earlier practice? (I don't mean just the scriptural allusions, but the use of a question-and-answer format when welcoming a new bishop.) What is the rationale for having the interview with a member of the laity?
I would like to try to address the second question, "What is the rationale for having the interview with a member of the laity?"

It does not seem clear from the ancient Canons that laypeople were explicitly called to approve and/or examine a candidate bishop.  It seems, though, that a bishop's election had to be approved by laypeople as well as clergy by the time of the 5th century, at least in some jurisdictions.  Leo the Great wrote:


  When therefore the choice of the chief priest is taken in hand, let
  him be preferred before all whom the unanimous consent of clergy and
  people demands, but if the votes chance to be divided between two
  persons, the judgment of the metropolitan should prefer him who is
  supported by the preponderance of votes and merits: only let no one be
  ordained against the express wishes of the place: lest a city should
  either despise or hate a bishop whom they did not choose, and
  lamentably fall away from religion because they have not been allowed
  to have when they wished.
  
  Letter XIV, To Anastasius, Bishop of Thessalonica, Chapter VI,
  "The Election of a Bishop must proceed by the wishes of the clergy and
  people."


There is perhaps a still better answer, but since the question has remained unanswered for two years, I thought I would at least contribute this.