Metafizzypop
Haven't looked it up, but as far as I remember, Picard associated the swords suit with water...
Oh, yes, I'd forgotten. In fact, I have the El Gran Tarot Esoterico that Abrac mentioned on page one here, and which also has Swords as water. So this element assignment would explain the water on this card.
But I'd still say the chessboard represents strategy. With water nearby, the card to me suggests a combination of both types of mental processes, of intuition and intellect, feeling and thinking, subconscious and conscious. There's a dualistic vibe to this card that I can't get away from.
ETA: I just took a look at my El Gran, and the Page of Swords shows a chessboard as well. It's looks like a big mat, and the Page is standing on top of it. No water is shown. The Page is standing mostly with his back to the viewer, and his hand is up on his forehead, above his eyes, as though shielding them, and looking out into the distance. He looks like he has his sights set on something. A plan? His enemy?