There's clearly no actual science behind the number, since time travel doesn't exist… yet. So was the writers' choice of that particular speed totally arbitrary?

Flip the number "8" on its side. Bam: that's the infinity symbol.

Yeah, it could just be a coincidence, but we suspect the writers were hinting at the infinite nature of time (and possibilities in history) here by making the speed at which time travel occurs a double infinity symbol.

We suppose they could have used just one infinity symbol, but it would have made those scenes so much less dramatic if Marty only had to get the DeLorean to 8 mph. At least he wouldn't have had to worry about harming any errant squirrels?

And there's another allusion contained in the number 88. There are eighty-eight keys on a piano, making it one of the most recognizably musical numbers around. How fitting that a music guy like Marty gets to travel at 88 mph back into the past—a past where, among other things, he introduces Chuck Berry to a little ditty named "Johnny Be Good."