Do you mind running a computer all the time and hack some code in?
You might try to find a way to acutate a little reed relay on beep/LED, maybe with a little help from a simple transistor.
(google for relay driver)
Then you could use serial port, even in the incarnation of a serial-to-usb-converter, hook up some of the handshake signals to your relay. This would signal the closure or opening of the relay to the pc.
(google for rs-232, handshake, dcd, dtr, dsr, rts, ...)
The PC could run a little program that waits for the handshake to change and display the time it happened.
(even a tiny python program would be sufficient.)