Is there a name for "the string of characters typed randomly on a keyboard home row"?

Solution 1:

You could say cat typing or the like, since it's a common meme to have cats walk on keyboards and typing.

Otherwise you can use asd or asdasd (Urban Dictionary) by antonomasia.

Some "dance" terms are also claimed to be in use, but I never encountered them.

Solution 2:

Button Mashing

This may be more of a gaming term but mashing the keyboard buttons is used as a phrase in other groups of people.

The act of rapidly pressing/hitting controller button(s) or (a) keyboard to play the game.

As always, there is an xkdc reference: this strip refers to 'fjafjkldsf' as "randomly keyboard mashing".