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I was checking the Hardware Guide for the Intel® Edison Compute Module (Section 4.4), and unfortunately the baud rate 31250 is not supported, you can use 38400 or 19200.
I already tested these values using the command: root@Leonardo:~# stty -F /dev/ttyMFD1 38400.
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Although the Edison UART doesn't officially support 31250, the data sheet that Leonardo refers to also claims that it is register compatible with a 16550.
Therefore, if you can find a way to write to the registers, you should be able to program it to just about any baud rate.
Alternatively you could patch the kernel and re-build it.