Can Blue-Eyes Shining Dragon be destroyed by Dark Hole in Yu-Gi-Oh?

Yes, it will be destroyed. "Blue-Eyes Shining Dragon" ("BESD") can negate the effects of a Spell, Trap, and Monster effects that designate/ targets it. "Dark Hole" does not target a specific monster so it can destroy "BESD".

An example of a card that designates is "Book of Moon", since the player chooses which monster to flip into face-down defence position. So "Book of Moon" could be negated by "BESD" and thus be unable to flip it face-down. If "BESD" were face-down (by perhaps the effect of a non-targeting card like "Swords of Concealing Light"), its effect would no longer be able to active and could be destroyed by effects that target like "Raigeki Break" or "Tribute to the Doomed".