I'll throw in my two cents worth on this one....
In its original inception, I doubt that anyone ever conceived of the J2 traveling at faster than light speeds - had they that capability then why put the Robinsons into cold storage?
Based on the "Alpha Control Manual", which has its flaws but is overall quite well done, the J2's flight to Alpha Centauri, a distance of 4.32 light years, would have occurred at relativistic speeds only, approximately 60% of light speed, which would have lasted 7.2 years. Allowing for time dilation at relativistic velocities, for the Robinsons, the trip would have seemed as if only 5.76 years... better, but still too long without the 'freezing tubes'.
Clearly, the 'hyperdrive' was a systems upgrade that was made to the J2 at some point after their initial launch, something that they picked up and jury-rigged along the way.
And, as long as we're speculating on that, then the inevitable questions arrives of 'how, at only 60% of light-speed, could they have become so lost so quickly?'.
Turns out, I've got an explanation for that (no surprise there)...
The first thing that happened to them once they awoke was encountering the 'derelict' craft, which essentially swallowed them whole. If you'll recall the scene where John Robinson is fiddling with an alien dial in what he concluded was their control room (a really stupid thing to do, under the circumstances), when he and Don West start seeing viewer images of different locations... Robinson presumes (in my opinion, incorrectly) the images to be some sort of star-chart, showing pictures of various nebulae and constellations and so forth. But, what if he were wrong?
Suppose, instead, that they were looking not at places to go but, as he turns the dial, they're looking at places that they now are? He could easily have been accessing the ship's jump-drive system and not have been aware of it. We never did see the outside of the derelict so, if we're imagining that the J2 could pick up hyper-drive along the way then its not too much more implausible to conclude that other races already have such technologies. By the time that they exited the derelict, they were hundreds and thousands of light years away from their original location. In essence, the reason that they were 'lost' is that they got that caused it themselves, even if only inadvertently.