**i) Definitions**:   
Let $F$ be a union closed family (U.C. family for short)     
 Let $S$ := $\cup_{ \Omega \in F} \Omega $ (the support of the family).  
 Let $F_{x} $ denote the members of $F$ containing $x$. 

**A) Union Closed Average variation **  : ( with $S$ being $T_0$-separated by $F$)

 $ \sum_{x \in S}{ |F_{x}|  }   is \ge |F|.|S|/2 $.  
( That is average $F_x$ size (on $S$) is greater than $|F|/2$)
 

**Note**: I cannot find any counter-example.  
The $T_0$ separation of $ S $ by $ F $ means that for any two different points ${x,y} $ there exist a member of $F$ containing one point and not the other(you can choose which!) .  

Another avenue of generalization that I cannot dismiss is a the multiset variation. 

**B) Mutltiset variation** :  
F is a family of $(\Omega,\eta_{\Omega})$  where:  
 - the $\Omega$ forms an U.C family  
 - $\eta$ takes its values in $\mathbb {N}^{\gt 0}$ 
  ( or $\mathbb {R^{\gt 0}}$ or $ [1,2,..,p]$ ordered naturally )   
 - for any $  \Omega_1 , \Omega_2$  of $F$ : 
 $ \eta_{\Omega_1 \cup \Omega_2} \ge Sup ( \eta_{\Omega_1} ,\eta_{\Omega_2} )$
  
Now the conjecture is the best measured $F_x$ is at least half that of $F$ where measure is the sum of members measure  : the standard U.C. conjecture being that of $\eta$ being the constant one function.