Heuristically determine a value f such that a probability d/f approaches 1/2
OK, another answer :)

Size of the set X' clearly depends on the distribution of points, choice of starting points, and random chance.

If the points are always chosen from the same distribution, I'd try drawing |X'| as a function of f and |X|. To do that, I'd just do some monte-carlo simulations.

Also notice that f is a linear function of mean distance between points.