Mark Bocko from Rochester University has done a proper analysis of this effect. Here's a summary:
The two interesting features of the sound at the end of the drop of the ice into the about 80m deep bore hole are the high-pitched sound (like a stereotypical sci-fi laser sound effect or a bullet ricocheting off metal) followed by low heartbeat-like pulses.
The "heartbeats" are essentially just echoes of the initial impact sound, and the spacing between them is the time it takes for the sound to travel down the hole, reflect at the bottom, and travel up again.
The high-pitched sound is due to the hole acting as an acoustic waveguide - think of it as a big organ pipe. It "conducts" frequencies along certain "modes", and the speed of sound of the excitations of these modes is dependent on the frequency in the higher modes - the higher the frequency, the closer the sound travels to the frequency-independent speed of sound of the ground mode.
So at the top of the hole, what we hear is first the ground mode + the higher frequencies from the higher modes, followed by the slower, lower frequencies in the higher modes, creating this high-pitched sound that gets progressively lower. The deeper the hole, the longer this "pitch-drop effect" will be drawn out in time.
For more details and some helpful pictures, you'll have to read Bocko's article yourself.