This is a pretty interesting topic, and one of the things that makes special relativity so mind blowing. But the answer is:
From your perspective on board the ship, the computer will function just as fast as it would if you were using it on Earth. After all, physics doesn't care about which inertial frame of reference you're in, so it wouldn't make any sense for the computer to work in some frames and not in others. However, from the outside perspective of a "stationary" observer, your computer would indeed be running slower -- though not at a standstill. Note: I put the word stationary in quotes because in special relativity, there's not really such a thing as stationary. All motion is, well, relative. What I should really say is "an observer moving quickly with respect to the spaceship", but that's not nearly as catchy.
These seeming paradoxes are explained away with the help of time dilation, length contraction, and the relativistic velocity addition formula. I might go more in depth here later if I have the time, but suffice it to say that the main place your thinking is going wrong is that you're unconsciously assuming that time and space are absolute. In reality, they start stretching and contracting when you move between different reference frames, and the end result is that the computer works just fine for the astronauts using it, but looks like it is running slow for outside observers. As an analogy, note that all biological processes take place at sub-light speeds, yet the astronauts will not perceive themselves as slowing down.
EDIT: Here's a slightly more math-y explanation, for those so inclined.
To truly channel the spirit of <PERSON>, rather than spaceships and electrons I'll talk about trains and baseballs. The concepts are exactly the same, but this is perhaps more concrete. 562
Anyway, suppose little <PERSON> is standing on top of a speeding train throwing his baseball, as kids are wont to do. Say the train is moving at $20 m/s$, and from <PERSON>'s point of view, he's throwing the baseball at $20 m/s$ in the direction the train is moving. Now, how fast will old man <PERSON>, who's standing on the platform, see the baseball travelling? Why, $40 m/s$, of course! How do we know? Well, we simply added the velocity of the train and the velocity <PERSON> throws the baseball at, since the baseball is already traveling at $20 m/s$ just by virtue of being on board the train.
Now, suppose the conductor got bored and told the engineers to speed the train up to $0.95c$, while <PERSON> undergoes intense physical training to get his fastball up to $0.95c$ as well. Now suppose <PERSON> has lived on this train his whole life. From <PERSON>'s point of view, he's just standing on a stationary train throwing a fast ball at $0.95c$. No biggie. From his point of view, the Earth is the thing moving really fast, while he's standing still and throwing the fastball at sub-light speeds. So there's no problem from his end.
But what about old man <PERSON>? Surely, he'd see the fastball moving at $0.95c+0.95c=1.9c$, violating the speed of light, right? Wrong! Due to some clever arguments (that actually only require basic trigonometry to follow, if you're interested), old man <PERSON> will actually see everything <PERSON> is doing being done slower than normal and squished along the direction of travel. The net effect is that to <PERSON>, the velocities of the baseball and train won't add in the usual way. Instead they'll follow the relativistic velocity addition formula: $$u = \frac{v+u'}{1+(vu'/c^2)}$$ where $u$ is the speed of the baseball old man <PERSON> measures, $v$ is the speed of the train, and $u'$ is the speed of the baseball measured by <PERSON>. Plugging in the numbers, we get $u=0.9986c$, rather than the $1.9c$ we would naively expect.
No go back and mentally replace every instance of the words "baseball" and "train" with "electron" and "spaceship", respectively, and I think you'll find the answer to your question. 844