11:11 wishes were not a 'thing' until the invention of flip-card (it's the wrong name, but the right function) clocks, with a rolling set of cards which would flip to the successive time. They are a predecessor of digital clocks.
What happened is the synchronization belief; most people do not take note of something which would otherwise be mundane, but they will take note of something which seems exceptional. 11:11 stands out, as it is a symmetrical time. How many times have you noticed that the time is 7:37 or 7:47 (similar to the Boeing aircraft), 9:11 (Americanized date format of the attack in New York in 2001), or any other number combination which may have significant meaning to you?
Well, somehow the time 11:11 was tied to a rumor about wishes, possibly for also being relatively close to midnight. Most children are sent to bed long before midnight, and would (their parents would hope) be asleep shortly after.
Children, in their purity of belief in so many things, would be so much more likely to believe that if they happened to wake up at 11:11 according to their digital clock, then a wish they made would come true. Some adults would believe the same.
Ultimately, however, it becomes a matter of belief. A magic spell, after all, is no more than an elaborate ritualized wish in most cases. It is investing a bit of energy into the hopes for a certain result.