I think Kishi hinted in some interview that the following could be possible:
If 3rd Hokage's (Hiruzen) wife had an affair with unnamed Hatake and so Jiraya was born, but then that unnamed Hatake guy got married and then Hatake Sakumo "White Fang" was born. And Jiraya could not be named after Hatake or Sarutobi so that nobody knows about the affair. Assuming Hiruzen's wife was from the same clan as he. So then Jiraya didn't have any last name.
And then later Jiraya married some Namikaze girl whom he met when travelling the world.'Cause there isn't any other Namikaze's in Hidden Leaf. And then she gave birth to Minato and so he got he's mother's last name and his fathers ninja genes.
And then Jirayas wife leaves him and he starts writing love novels. Then he's brother White Fang starts reading them at the time he got depressed. And likes them a lot. That's why Kakashi also likes them. But then one day he criticises Jiraya of he's newest novel and then Jiraya kills him. So the suicide was a cover up that Jiraya made.
Later Jiraya regrets that he killed Sakumo, so he suggests that his son now the great Yellow Flash would teach Sakumo's son Kakashi and teach him the Rasengan so that he won't be sad that his dad died.What do you think?