Let me point out that in the case $X=\mathbb{N}$, the assertion seems to be simply equivalent to the existence of an injection $\omega_1\to \mathbb{R}$. Ricky has cleverly proved the forward implication. But conversely, if there is a such an injection of $\omega_1\to\mathbb{R}$, then we can define $d_X(i)$ to be the first real not in $i(X)$.

This idea generalizes to higher well-ordered cardinals as well. If $X=\kappa$, then the existence of a map as you request is equivalent to the existence of an injection $\kappa^+\to P(\kappa)$. Ricky's argument again gives the forward direction, and conversely, if there is an injection $\kappa^+\to P(\kappa)$, then we can let $d_X(i)$ be the first set on the list not in $i(X)$. 

Perhaps it is true that for any set $X$, the property on $X$ is equivalent to the assertion that the Hartog number $\aleph(X)$, the first ordinal not embedding in $X$, injects into $P(X)$. The converse direction is the same as above, but it isn't clear to me whether one can push Ricky's argument through for this.