Your question is about what will be the speed of elektron? Is this sum of speeds of electron and the ship, or not? If it is a sum of these speeds, speed of elektron will be more than speed of light ? If not, then there will be a time which is slowing down, so,"can the electron reach to command in circuits? Is not it?
Firstly the speed of electron won't be more than speed of light acccording to passenger. It will be as usual for passenger. Acoording to observer, if possible, the speed of electron will decelarate by the slowing time and sum of the speeds of ship and electons will be again same as usual. So your computer will work properly.
According to my view, in reality, the time inside ship is under different physics rules. Because by giving more energy to accelarate the ship with it's increasing mass, (I'm thinking according to String theory), the strings are energized and extended, so they can compress the spatial dimensions (on the same route with accelaration). Also "the time" is a wave kind, sourced from universal expansion (it is again my idea, not a proved scientifical data). So time of the ship will be acting as waves. And you know the behaviour of waves while they are passing from different environment as usual. Their wave lenght and speeds are changing, after left this environment, they turn again their ordinary position. So, if you think "the time" as waves, which are also in the speed of light, limit of speed is the limit of these waves. And, if an object is accelarated in these waves, it will be limited at doppler barrier. Because all waves will be overlapping. (And their energy will transfer to mass)