Recently I saw the instructable here by gregoryfenton about fixing the serial port on the LinkIt ONE. That was great but I would like to provide an alternate method which I believe is better.
First, to clear up any other issues, make sure before plugging your board in, that the MS/UART switch is set to UART. You will not be able to get the serial monitor open without the switch set to UART.
Once you have that switch set proceed to the next step.
Step 1: Correct COM Port
This step is easy to miss, but when you plug in your board and go to Tools > Port there will be 2 ports. One is serial and the other is not. You must check both of them, by trying each separately until one works. Once you have found the right one, just select that one each time you plug in the board and you will have serial.
However, sometimes it will not print out the first few lines you want because it has not yet initialized. Proceed to the next step.
Step 2: Wait for Serial Monitor
Sometimes the board will not print the first messages you want it to print because it is not fully initialized. To fix this simply add these two lines it your Setup function:
The reason this happens is because the LinkIt ONE is similar to the Arduino Leonardo in that the COM port is emulated by USBCDC. If it is reset the COM port connection will disappear.