"Hey Jude" was specially written by Paul for Julian Lennon, the oldest son of fellow Beatle John Lennon, to comfort the little boy over his parents' messy divorce and his father's constant absence and emotional neglect.
Paul: I started with the idea ‘Hey Jules’ –- which was Julian –- ‘don’t make it bad, take a sad song and make it better. Hey, try and deal with this terrible thing. I knew it was not going to be easy for him. I always feel sorry for kids in divorces... I had the idea [for the song] by the time I got there. I changed it to ‘Jude’ because I thought that sounded a bit better.
When asked about his childhood, Julian has been quite open about Paul being more of a father to him than John ever was. And aside from some ups and downs, they've apparently remained close well into Julian's adulthood.
Julian: Paul and I used to hang about quite a bit... more than Dad and I did. We had a great friendship going, and there seems to be far more pictures of me and Paul playing together at that age than there are pictures of me and my dad.