First and foremost, major props to Optionparty and John U who pretty much already nailed it.
The welding action itself is a brutally noisy high amperage DC current applied to the body or frame of the vehicle. In a perfect world, the battery would be disconnect, no current would be flowing in its systems, and the welder would have an extremely short path to its return ground, with no stray currents. In the real world, the battery is in, the car is on, and the ground lead stays in one spot on the other side of the vehicle while the welder goes all over the place.
Since the vehicle's body and frame is the ground reference for the onboard systems, its pretty conductive, and the potential for EM broadcast is massive. This high frequency broadcast could scramble control signals in computers by induction directly over those signals, or by inducing noise between the input power line and ground. A Faraday cage around the device could very well be a solution, except that you don't have access to an actual earth ground, and you don't want to reference it to the car's ground, because, well, welder. It's not a problem, it's a consideration - you have to leave the Faraday cage floating, which means any received EM will be rebroadcast on the other side of the cage.
Radio frequency chokes on input leads will prevent anything wild from getting into the power supply circuit for your electronics. Shunt capacitors should also help, but I would put them before the choke, in the event a stray current decides to take a path up the shunt cap into the power line - it shouldn't happen but, hey, the battery should have been disconnected, right?
If your device takes in small gauge instrumentation signals from other places, you probably want to do two things. One, specify shielded wiring. And two, isolate that signal from the actual receiver, using a capacitive or optical isolation IC. The shielding is a bit double edged. You want it to protect the inbound signal from high frequency corruption and cross talk, but now it stands a chance of becoming a stray path. The isolation can protect you to something like 2500V, pretty cheaply. So hopefully the worst case scenario now is that the input becomes incomprehensible garbage, which the circuit should be reasonably ready to reject.
And fuses. Don't forget fuses. Fuses are your friend. Or resettable polyfuses, if appropriate.