Try filtering out the rows that contain strings with the delimiter and work on those only like:
SELECT SUBSTRING(myColumn, 1, CHARINDEX('/', myColumn)-1) AS FirstName,
SUBSTRING(myColumn, CHARINDEX('/', myColumn) + 1, 1000) AS LastName
FROM MyTable
WHERE CHARINDEX('/', myColumn) > 0
Or
SELECT SUBSTRING(myColumn, 1, CHARINDEX('/', myColumn)-1) AS FirstName,
SUBSTRING(myColumn, CHARINDEX('/', myColumn) + 1, 1000) AS LastName
FROM MyTable
WHERE myColumn LIKE '%/%'