The cotton gin, while invented in 1793, did not catch on in popularity until around 1815, at which time the demand for slaves and the number of cotton plantations rapidly increased. This made cotton cheaper and more plentiful, and soon after, the American textile factories began to appear in New England. So I could argue that early industrialization in the North, and the growth in wealth, population and cities that followed, was directly related to the cotton gin.
In general, the Northern economy was very involved with slavery. Therefore, the invention of the cotton gin did have an impact on the North.
The North provided many services that were needed by slaveowners in the South. For example, it was Northern companies that tended to insure slaves or to loan money to slaveowners in the South. The North also tended to sell many kinds of goods to the South since the South did not have many factories of any sort. There was a joke that said that, at a Southern funeral, the South provided only the corpse and the ground because the coffin and its nails and the shroud for the deceased and everything else was made in the North. In addition least some of the cotton grown in the South was made into textiles in Northern factories. All of these types of business grew after the cotton gin was invented because of the increase in the number of slaves.