Does anybody know why it had a launch window of only one second?
Probably because the relative speeds are so fast, that if they launched it even seconds later they'd need to plot a whole new course. The comet is moving at a relative speed of something like 23,000 mph.
The only reason I can think of is that it's a really performance-limited mission, on the part of both the booster (get DI as high as possible, as accurately as possible), and Deep Impact (it takes propellant to account for the orbit plane changes associated with wider launch windows).
Note, BTW, that a "one-second launch window" doesn't really make exist. IIRC, the Delta launch sequencer works on the even second. So basically, they said: launch at exactly this time.