According to a line cut from the filmed episode, Dr. Crusher informs the three revived patients that "Nurse Wright will furnish you with anything you need." 
In 2365, the Enterprise began experiencing problems with its subsystems as result of an Iconian software transmission that had affected the ship. Wright got chewed out by Dr. Pulaski who was complaining that the biobeds were not working, the ship was falling apart, she had 35 emergency calls scattered across 12 decks and her trauma teams were being run ragged trying to respond. As she stormed away, Wright walked off in another direction shaking his head. (TNG: "Contagion")
Wright assisted Dr. Pulaski again when a medical team was ordered to the shuttlebay and discovered an unconscious double of Captain Jean-Luc Picard on board the shuttleEl-Baz. (TNG: "Time Squared")
In 2366, Wright was transferred to the command division and promoted to an ensign where he manned the conn when the nanites took control of the Enterprise-D. He was present when the bridge was flooded with nitrogen oxide. Later his console exploded but Wright was not harmed. (TNG: "Evolution")
Their name originated from a scripted source. Wright, who was portrayed by recurring background actor David Eum, was identified in the "The Neutral Zone" script as "Nurse Wright".  In the script for "Contagion", Wright was referred to as "one of [Pulaski's] medical staff" and "the chastised nurse."