What if Bad Luck Brian invented a time machine?

It may have just been a typo or you forgot something, but in that last part, I think you meant to say he would have been born. (Dad is unable to go back, kid is conceived, born, and grows up to make time machine, which restarts the loop.) This type of paradox is known as the "Grandfather Paradox" because most people are familiar with the paradox where you kill your own grandfather then you can't be born and you can't kill your grandfather so you get born then repeat. There are two main types of paradoxes, Grandfather Paradox and the Origin Paradox, which (as you're a whovian, I'll use this reference) is how River Song was named after herself.

Yes, that's what I meant to say. Thanks for clearing that up for me. (And I like the Doctor Who reference :) )