Why do some board games have long-lasting appeal?
I agree with Lo'oris' argument that most older well known games are still widespread mostly because they already are everywhere. How many people go out and buy chess boards compared to the number of people who simply have them around from previous generations?

I think there are some additional aspects:

Simplicity

In my experience games need a good balance between complexity and depth in order to be popular. Basically the ratio of options a player has divided by the amount of rules he has to learn. A lot of games that are popular in our family are relatively easy to learn but offer a lot of possibilities in comparison. This can be illustrated especially well with games that are in very similar but somehow one of them just feels better. Compare Pachisi and Dog/Tock. The have the exact same basic concept — move your pieces around the board one time and get into the safe haven. But in all the groups I've played in Dog/Tock has completely displaced all other Pachisi variants because with a few additional rules it has become a much more interesting game.

Uniqueness

Most modern boardgames aim to produce a specific atmosphere or experience that is perhaps unusual for the players — a feeling they would otherwise not have. With so many games out there a lot of times games can feel similar to each other. There are countless games where you throw dice, move along a path and do something according to the field you landed on — nothing special. And then there are games that somehow manage to capture something special. For example Pandemic expertly captures the rush of working together against an overwhelming threat, Bohnanza produces an intense trading atmosphere, and games like Werewolves or Battlestar Galactica allow people to play out their mischievous and paranoid traits.

Whenever a game is released that succeeds with a good combination of these two (and maybe other) aspects then it is likely to become quite popular. Of course others will try to copy and improve that mixture. Some will fail but others will eventually succeed. At that point a battle will take place between the improved qualities of the newer game and the established following of the older one.