According to wiki and the "Forbes Fictional 15", Gomez Addams (originally penned in the comics with no surname, the name Gomez was chosen by John Astin, the actor portraying him for TV), the source is given as inheritance and investments.
The character is purported to be descended from Castilian royalty and British aristocracy, which would account for the inheritance portion, and the other comes from lucky (Rather than sage) investing. One example given is a swamp that he bought for "scenic value" that turned out to have massive oil deposits, another is a mummified hand he bought at a flea market that turned out to be from an Egyptian pharaoh.
According to the Forbes article, his net worth is guessed at ~ 2 billion dollars US. It is to be noted, that due to Fester gaining control of the fortune, Gomez dropped off the fictional 15 list in 2008, and does not appear to have returned, despite having a presumed worth of more than some of the lower members of the list in the 1 billion US range.