In Catholicism (and indeed many other branches of Christianity), the six weeks leading up to Easter are a time of contemplation and purification before the major festival. This period is known as Lent, and it is common to deny oneself something (say, sweets, chocolate, or playing computer games) as part of one's preparations (or at least it used to be — these days there is a strong trend towards doing something positive such as doing something for charity).
Thus in Catholic households, at a certain time of year, it's not uncommon to hear declarations such as "I'm giving up [chocolates] for Lent".
The joke puns on the similarity in sound between Lent and lint, a word that can be used among other things to describe the fuzzy accumulation of clothes fibres that sometimes builds up in one's navel: he's saying he's giving up picking the lint out of his bellybutton, but in such a way that it sounds like he's giving up picking his bellybutton for Lent.
For this he deserves an almighty thwack.