By misspelled family names, I assume you mean misspelled surnames.
Myself, I've stopped thinking of them as misspelled, and instead as alternative spellings of the same or similar name. My own simple one-syllable surname, I have found in various US, German, and Lithuanian records as Lak, Lack, Lok, Lakas, Liak, Lyak, Locke, and Lock, all based on the Yiddish surname 'Lamed-Alelph-Koph' (from grandfather's US tombstone). I do consider a misspelling to be when a US census record has the surname as 'Lech', though perhaps the census taker mis-heard it.
Looking for ship passenger lists was the most challenging, until I realized that a clerk in a shipping company in Hamburg or Bremen would write down the name with German spelling, which would be Lak or Lack. I searched for both these spellings then, and also tried my luck with Lok. Was a lot more fruitful.
Using the SteveMorse.org option of 'Starts with' was also helpful, for both first names and surnames, along with the approximate birthyear (+/- 5 years minimum) and year of arrival (+/- 3 years minimum). I have had more luck doing broad searches rather than focused ones, especially with immigrants mis-remembering their year of arrival, and having only the broadest idea of what year they were born.