Never Kill A Boy On The First Date - S1-E5
Deliberate mistake: When Buffy and Giles are searching the funeral home, we see three shots of Buffy and Giles opening up different mortuary-cabinet doors, shot from the point of view of the cabinet. The problem? You can tell they're opening and closing the same door over and over again. The background is exactly the same, and all they've done is slightly re-arrange the stuff behind them, and had the actors come in from different angles to create the illusion they're opening different doors. It's SUPER obvious.
Chosen answer: From what we see in the series, certain girls are identifiable as potential Slayers - Kendra clearly fell into this category. The identification method is presumably mystical in nature, but the Watchers' Council are pretty effective at that sort of thing, so they're quite good at tracking down the potentials ahead of time. Not perfect, though - it does appear that Buffy herself may have slipped through the net - certainly she had no inkling of what she was until she'd already taken on the role of Slayer. It is possible, however, that this was actually cultural - an American family would hardly be likely to turn over their daughter to some strange man for 'training', so the Watchers might have chosen to keep an eye on her covertly, whereas some other cultures (like Kendra's Jamaican parents) might be more willing to believe.
Tailkinker ★