What is the correct syntax for a set command within a mySQL user defined function?

you can use CASE...THEN...ELSE block

DELIMITER $$  

CREATE FUNCTION removeMethodAndBackslash(input VARCHAR(40))
RETURNS VARCHAR(20)
    BEGIN

    DECLARE loginName VARCHAR(40);

    SET loginName = 
       CASE WHEN SUBSTRING(input,7) LIKE '\b%' THEN 'b' + SUBSTRING(input,8)
            WHEN SUBSTRING(input,7) LIKE '\n%' THEN 'n' + SUBSTRING(input,8)
            WHEN SUBSTRING(input,7) LIKE '\t%' THEN 't' + SUBSTRING(input,8)
            WHEN SUBSTRING(input,7) LIKE '\r%' THEN 'r' + SUBSTRING(input,8)
            ELSE SUBSTRING(input,7)
       END;

    RETURN loginName;

END $$

Try this one -

DELIMITER $$  

CREATE
    FUNCTION removeMethodAndBackslash(input VARCHAR(40))
    RETURNS VARCHAR(20)
    BEGIN

    DECLARE loginName VARCHAR(40);
    SET loginName = input;

    IF SUBSTRING(loginName,7) LIKE '\b%' THEN
        SET loginName = 'b' + SUBSTRING(input,8);

    ELSEIF SUBSTRING(loginName,7) LIKE '\n%' THEN
        SET loginName = 'n' + SUBSTRING(input,8);

    ELSEIF SUBSTRING(loginName,7) LIKE '\t%' THEN
        SET loginName = 't' + SUBSTRING(input,8);

    ELSEIF SUBSTRING(loginName,7) LIKE '\r%' THEN
        SET loginName = 'r' + SUBSTRING(input,8);

    ELSE
        SET loginName = SUBSTRING(input,7);
    END IF;

    RETURN loginName;

END $$

DELIMITER ;