In the The Flash episode The Nuclear Man, we see Barry and Linda making out, and he vibrates. That brought the question to my mind. If Barry ever decides to have a kid, would it also have super speed because Barry's DNA has mutated and we tend to pass DNA to future generations?
It depends on whether or not the mutation is "germ line". Each cell in your body has its own set of genes, and if a cell in your eye has a mutation, this won't be passed on to any children.
This is actually sort of a big deal, as medicine takes the first steps towards being able to cure genetic diseases. They have methods of altering the genes inside your cells such that if you had cystic fibrosis they might be able to force the cells of your lungs into behaving properly. However, medical ethicists are reluctant to approve any such therapy if there is the potential for the changes to be passed on to children. (Thankfully for cystic fibrosis sufferers, no one procreates with their lungs.)
If you follow the news, this subject shows up from time to time.
Yes, Berry's genes will pass into his son and daughter. Twins actually.
From the comics we know his son Don and daughter Dawn will get fraction of his powers.
See this Don Allen (New Earth)
Don Allen was the son of Barry and Iris Allen, conceived in the 30th Century after his father's retirement, but born after his death in the first Crisis. He and his twin sister Dawn inherited a fraction of his speed
Similarly true for the daughter Dawn Allen (New Earth)
You can see the Family Tree
They are the Tornado twins
Dawn and her twin brother Don were born in the 30th century. Both twins inherited the power of superspeed from their father, Barry Allen. Despite widespread fear and hatred towards metahumans and their powers in the 30th century, Dawn and Don chose to do good as superheroes. Using their superspeed to spin around quickly, they appeared as tornadoes and were known as the Tornado Twins.
Now if the tv show maintains these comics then we will see this genetic transfer in future.