I asked why the speed was 88 MPH, and it made a lot of sense.
Now I want to ask: Why does the Flux Capacitor need 1.21 jigowatts to travel in time? Is this a magic number?
Lightning strikes deposit somewhere on the order of a few Mega-Joules, not enough to power a flux capacitor, but it seems that the peak power of a bolt is about 1 terrawatt, a thousand times too large; you'd fry the delorean with that sort of energy.
It appears that 1.21 Gigawatts is roughly the energy of a nuclear power unit, though this sort of unit would be made of several rods, many more than those that fuel the Flux Capacitor.
Unless we find the dubious science correspondent referred to by the directors in the commentary (the one who pronounced it Jigowatt) we'll never know if they advised this number or it was plucked out for 'sounding nice'.
Bob Gale (the film's creator) spoke to this in a recent interview. In short, the number "sounded good" and was evidently based on a brief conversation he had with an electrical engineer working as a consultant for the film about the amount of energy contained in a bolt of lightning.
MTV: OK, let's geek out for a minute. 1.21 giggawats. Why 1.21? Why not 1.22 or 2.21?
Gale: We did some research, and I think it's relatively accurate. 1.21 sounded good. It is like, why 88 miles per hour? It's easy to remember. As a writer, you want to find the words that'll sound right so it'll be stuck in the audience's mind. The electrical engineer we spoke with about how much electricity is in a lightning bolt, he pronounced it jig-a-watt. We'd never heard the term before. You can call it gig-a-watt or jig-a-watt. They're both correct. I actually misspelled it in the script, and spelled it jigowatt. That's why Doc and Marty pronounce it that way.
'BACK TO THE FUTURE' CREATOR BOB GALE ON EASTER EGGS, ERIC STOLTZ, SEQUELS AND THE 25TH ANNIVERSARY
(transcription errors corrected by me)
It is a large number, one which defies non-nuclear methods of portable generation (even to this day).
The number has some appeal visually, and could be spoken memorably by the actors involved.
It perfectly fit the story purposes, and is not outside of the bounds for a lightning strike, as Pureferret says.
A number low in the gigawatt range was exactly what was needed, and Doc Brown was equal parts crazy and brilliant - he would have calculated how much power is needed, and would also use the exact number (1.21) instead of rounding it as most people would. Thus, 'one point twenty-one' - it's a good number to speak, it doesn't have too many syllables. Most likely, one of the writers picked it at random (possibly along with several others) and the writers decided it fit best.
To answer your direct questions its a MacGuffin just something to advance the plot.
How to answer the inferred question of why the need for so much power (in this case plutonium/Mr. Fusion/Steam which in fact is prove of the MacGuffin) and the speech of 88MPH is this:
The speed is required seemingly to provide a object with the forward motion need to start a time dilation (i.e why traveling in a airplane you seem to be barely moving while someone standing on the ground sees a very fast plane). The power is a requirement to generate a temporal flux to propel the object at the cusp of a time dilatation to another point in time.
To quote Remiel:
[...] you CAN generate 1.2 GW pulses off of a lower average-power supply [...] [...] you'd need a ridiculously awesome capacitor to do that [...]
and that is also why that is called flux capacitor. :)
To answer the question I think they chose 1.21 Gigawatt because it sounds good and it looked like a very big number at that time.