I think the problem is not in the cable.
Though I am not using the same cable with you, I can tell you how I check the cable and board:
1. Cable check:
Connect 3.5mm to rs232, and rs232 to usb, and usb to my laptop. Now I can see a COM device appears on my laptop. (I don't connect the board right now because we just want to check the cable). Next, I use a small wire to short the RING and TIP of 3.5mm audio port. We know the RING and TIP are connecting the RX and TX of serial port, so we actually connect the RX and TX(the 2 and 3 pin of serial port), and what we input will instantly appear on the putty session.
The result is exactly what we expected. The input is exactly the output, no lag, no error, very reliable. It means that the 3.5mm to serial cable, the serial to usb cable and the usb cable driver are all working correctly.
2. Galileo board function check:
I boot with both the image in SPI flash and full Linux image in SD card. (SD card image are downloaded directly from Intel website). Both can work correctly, and I can upload sketch through USB client port. I can also connect the board with ssh and telnet, with no lag. The only thing that not works correctly is the audio jack serial port.
Since your board's audio jack is good. Could you use a multimeter and measure the resistance between the TIP TXD and SLEEVE gound on your board?
I want to know if the TIP TXD is open to SLEEVE gound on your board, thank you!