I like the question. You are actually asking, what is the smallest finite probability space (in size of $\Omega'$) on which one can have $d$ distinct pair-wisely independent (but not necessarily independent) events of probability $\frac{1}{2}$. Just to explain the reformulation: once the $i$-th event is defined to be
$$E_{i}=\lbrace\omega\in\Omega'\;|\;\omega(i)=1\rbrace,$$
the "efficient substitute" criterion amounts to $\mathrm{Pr}(E_{i})=\frac{1}{2}$ and $\mathrm{Pr}(E_{i}\cap E_{j})=\frac{1}{4}$ for every $i,j$ with $1\le i,j\le d$ and $i\not=j$. The example for $d=3$ is a [well known case][1] of this situation.

Now, every such space satisfies $|\Omega'|=4n$ for some natural $n$ (the reason being obvious). Knowing this, we could ask an inverse question: given a $4n$-set $\Omega'$, what is the maximum size of a family $\mathcal{F}$ of $2n$-subsets of $\Omega'$ such that every two of them have intersection of size $n$. Knowing a precise answer to this, the original problem is solved as well: given a $d$ take the smallest $4n$ such that the maximum size of $\mathcal{F}$ is at least $d$.

There is a [paper][2] titled "Pairwise intersections and forbidden configurations". Let me quote from the abstract:

> Let $f_{m}(a,b,c,d)$ denote the maximum size of the family $\mathcal{F}$ of subsets of an $m$-element set for which there is **no** pair of subsets $A,B\in\mathcal{F}$ with $|A\cap B|\ge a$, $|A^{c}\cap B|\ge b$, $|A\cap B^{c}|\ge c$, and $|A^{c}\cap B^{c}|\ge d$.

The count we are looking for is exactly $f_{4n}(n+1,n+1,n+1,n+1)$. Besides other interesting things, the paper gives also asymptotic estimates for this count; one of them gives $\Theta(4n^{2n-1})$ in our case.

Edit: unfortunately, I was too quick with the asymptotic estimate. The paper gives an estimate only for fixed $a,b,c,d$ as $m$ tends to $\infty$.

  [1]: http://en.wikipedia.org/wiki/Pairwise_independence
  [2]: http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WDY-4K5SSV9-2&_user=10&_coverDate=11%252F30%252F2006&_alid=1751979224&_rdoc=3&_fmt=high&_orig=search&_origin=search&_zone=rslt_list_item&_cdi=6779&_sort=r&_st=13&_docanchor=&view=c&_ct=11541&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=f7d789c702d869ebb39ba6aed9d790e2&searchtype=a