What is a good notation for the 'set' (or stack if you insist) of all principal G bundles over 'all' spaces for given G? BG is way over used. How about Bun(G)?

In my comment to the question, I may have misinterpreted Jim's intent. Upon a second reading, I think Jim would like to consider the collection of all pairs consisting of a space $X$ and a $G$bundle over $X$. If this is the correct interpretation, then I think the correct notation for this category is $\text{Spaces}_{/\mathrm{B}G}$. The meaning of this notation is: $\mathrm{B}G = \operatorname{Bun}_G$ is the stack of $G$bundles, defined so that a map $X \to \mathrm{B}G$ is a $G$bundle over $X$. The category $\text{Spaces}$ is whatever category of spaces you want to consider (topological spaces, manifolds, schemes, ...). The category of pairs $(X,$ $G$bundle over $X)$ is the category of pairs $(X,f: X \to \mathrm{B}G)$, which is precisely the "comma" category of "spaces over $\mathrm{B}G$", and that's what's notated by the subscript. 


In algebraic geometry, $BG$ only ever refers to a stack, viewed as a fibered category over the category of schemes, or affine schemes, depending on your conventions. The objects are principal $G$bundles $P \to X$ of schemes (or if $G$ is a group sheaf, it is a map of sheaves). The morphisms are $G$equivariant fiber squares. More to the point, there is no alternative object in the algebrogeometric universe that takes the name $BG$, so there is no source of confusion there. The name $\operatorname{Bun}_G$ is usually reserved for the enriched Homstack construction $\underline{\operatorname{Hom}}(, BG)$. In other words, $\operatorname{Bun}_G(X)(T) = BG(X \times T)$. 

