Today, I heard that people think that if you can prove the Hodge conjecture for abelian varieties, then it should be true in general. Apparently this case is important enough (and hard enough) that Weil wrote up some families of abelian 4-folds that were potential counterexamples to the Hodge conjecture, but I've never heard of another potential counterexample.
Anyway, in short:
1) Does the Hodge Conjecture for abelian varieties imply the full Hodge conjecture?
2) If not, is there an intuitive reason why abelian varieties should be the hardest case?