What is the first name of Leonard's mother?

The first name of Leonard's mother is Beverly: she is Dr. Beverly Hofstadter.

Beverly is one of the most psychologically influential members of Leonard's family, even though she appears only as a recurring character. She is a psychiatrist and neuroscientist whose clinical, analytical approach shapes the way she treats her son. Leonard often feels that she studied him as a subject rather than offering the warmth and reassurance he wanted from a parent.

Christine Baranski played Beverly throughout her appearances on The Big Bang Theory. The character first arrives in the second-season episode “The Maternal Capacitance,” immediately forming an unexpectedly close intellectual connection with Sheldon. Her similarity to Sheldon—especially her rigid logic and limited emotional sensitivity—is a recurring source of comedy.

Beverly is sometimes confused with Mary Cooper, Sheldon's devout mother, or Debbie Wolowitz, Howard's overprotective mother. Beverly is Leonard's mother, while Leonard's father is the anthropologist Alfred Hofstadter. In the final season, Leonard confronts Beverly about their painful relationship, and she finally embraces him.