You can build a simple recording device from a PIC. Use an electric condenser microphone (ECM) with the correct preamp (can be built from a few transistors, resistors and capacitors) to get the output voltage within 0-5V. Then feed it into the PIC's analog-to-digital converter (ADC) unit, which will convert the voltage (0-5V) to a byte value (0-255). You can then transmit the bytes to the computer serial port via the PIC UART unit. A simple software running on the PC (something like Hyperterminal) will save those bytes into a file.
If the PIC has some storage capability such as external EEPROM or SD-card, you can also save the data without needing for a PC client.
Given the PIC and the serial speed, you can probably record at 8bit, 8kHz sampling rate. The analysis of when the beep happens will probably have to be done on the computer side, as the PIC is nowhere fast enough for audio recording and processing at the same time.