“Herbert introduced me to his father, who lived the other side of London, in Hammersmith.”(GREAT EXPECTATIONS by Charles Dickens–retold by Clare West)
There is not any preposition before “the other side.” Is this a usual expression?
In speaking, yes. In writing I might say “lived in Hammersmith, ON the other side of London”
Yes that’s a usual expression, especially for Londoners. Come to think of it, I can’t really think of any preposition that could be inserted into that sentence to make it sound and look ‘right’. Maybe the word ‘across’ to make it ‘who lived across the other side’ but still doesn’t exactly sound right. The word order could also always be changed, to make it ‘who lived in Hammersmith, the other side of London’.
Beaten to it 
Can you say “I live the other side of London?”
I think most English speakers woldwide would say “I live ON the other side of London” is standard. Dropping the “on” is dialect. Another example is “I went down TO the pub.” (worldwide) versus “I went down the pub.” (England)
I wonder if we can compare “the other side” with “the other day.”
You can… I am going down the pub the other side of London 
Thank you for replying to my question.
You’re welcome.
As long as we don’t live the wrong side of the track we can all go down the pub the other side of town. It’s very English, as you, Dooo, said.