What's the second non-repeating character?

MATL, 11 bytes

tk&=s1=)FT)

This exits with an error (allowed by default) if there is no second non-repeated character.

Try it online!

Explanation

t      % Implicitly take input string. Duplicate
k      % Convert to lowercase
&=     % 2D array of equality comparisons
s      % Sum of each column
1=     % True for entries that equal 1
)      % Apply logical index to the input string to keep non-repeated characters
TF)    % Apply logical index to take 2nd element if it exists. Implicitly display 

Retina, 25 bytes

i!2=`(.)(?<!\1.+)(?!.*\1)

Try it online! (The first line enables running the code on a test suite of several inputs.)

Explanation

This is just a single regex match, the regex being:

(.)(?<!\1.+)(?!.*\1)

That is, match a character and ensure it doesn't appear anywhere else in the input. The rest is configuration:

• i activates case insensitivity.
• ! tells Retina to print the matches as opposed to counting them.
• 2= tells Retina to print only the second match as opposed to all of them.

05AB1E, 15 12 bytes

l©v®y¢iy}}1@

Explained

l©            # store lower case string in register
  v     }     # for each char in lower case string
   ®y¢iy      # if it occurs once in string, push it to stack
         }    # end if
          1@  # push the 2nd element from stack and implicitly display

Try it online

Saved 3 bytes thanks to @Adnan