Ben was born when the thing that dispenses soda in Bev, dispensed him (Philip J. Fry I nearly drank him!)
When Ben was born, Bender wanted nothing to do with him, so he gave him to Bev. Bev refuses, so Bender reluctantly takes care of Ben. But he later takes a shining to Ben.
Ben was named after the first three letters of his father, to whom he has a strong resemblance. He wanted to bend just like his father who raised him, however, Bender said his family's bending abilities are passed down through the mothers, but his mother Bev was a soda machine, thus she didn't have any arms, so he couldn't bend and he can't have a bending card chipped in him, because there's only one slot in him, which holds his memory. After winning custody of Ben, Bender decided to help Ben learn how to bend, by allowing Professor Hubert J. Farnsworth to install a bending card, at the expense of all of Ben's memories.