Jeff Whitley gave Royle`s men the lead, but Duncan Ferguson equalised before Weaver became the main focus of attention.
Shortly before the interval, he had a rush of blood - racing off his line and hauling down Kevin Campbell to give The Toffees a penalty.
Weaver nearly atoned for his error, saving Michael Ball`s spot-kick, but the defender bundled home the rebound.
And, as the visitors pressed for an equaliser, Weaver allowed David Weir`s header to slip through his grasp and gift their relegation rivals all the points.
In between, though, Weaver made a string of fine saves, denying Steve Watson, Idan Tal and David Unsworth, but continued blunders could cost the team their Premiership status.
Despite this, Royle refused to condemn the 22-year-old, saying: "Nicky`s misfortune was symptomatic of his and our whole season.
"We started well, passed it well and made chances, but we keep shooting ourselves in the foot.
"Nicky has done very well, and I was more concerned about the free header than the bounce that led to the goal.
"I will stick by him because he is an excellent keeper."
Weaver tried to defend himself, pleading for the opportunity to prove his talents, saying: "I still have faith in my ability. But when you make mistakes, you have to live with them.
"The manager has backed me so far and, as long as he continues to pick me, I`ll be happy.
"This hasn`t dented my confidence and I hope to stay in the team to help us fight relegation."
