You're barking up the wrong tree. This has nothing to do with meditation aids.
This is about a user who already enabled a "distraction free" time on their phone, and how you might design a bypassing process which (A) deters a user who was really just looking for a quick dopamine hit and (B) tries to let them through in a way which is less-likely to lead to immediate backsliding.
I only mentioned meditation because it would be nice if a hypothetical psychic mind-reading phone could tell that you were about to use it mindfully.
> You want people to stare at a blank screen as a form of introspection.
No, the real point is that it is boring and imposes a delay before gratification. Any kind of boredom-driven introspection is a side-benefit.
> Good lord you think that task is relatively easy?
Remembering three digits while having no other demands on your decision-making nor on your memory? Yes, you ought to succeed at that.
> Also what is this "uncomfortable alone with their thoughts time"?
"Hey, I've been staring at this goddamn thing for what feels like ten minutes already... did I really need to bypass this lockout just to check if I got another e-mail or tweetstagram?"