It's really an interesting project. I have also a 8840A with GPIB and TRMS options. Unfortunately I have only PCI GPIB card and I use a laptop usually. It would be nice if I could use my laptop with this puppy. Of course I can build a USB-GPIB thingy but this is also interesting. Maybe I can help with some information how the communication is going.
First of all the Instruction manual contains the schematic of the option board. The funny thing is that I have an another revision of the board with different ICs and layout but at least the connector pinout is the same
You can use this schematic to create your own differential transmitter/receiver.
I'll try to sniff around with a TTL-RS232 converter in the protocol in the coming
days weeks. I hope it is a human readable format. If it is I can find out the baud rate by trying. Or if someone has the option board and a digital scope please catch a transmission to measure the baud rate.
Arduino may not be the best option. The ATmega328P has only one USART and it is used for the USB communication too. From software point of view it is easier to use a micro with two UARTs than write a software UART.
I've attached the schematic and two picture from my board. Maybe they can help somehow. WARNING! It's not the same as on the schematic!