The `set' of all principal G bundles over `all' spaces 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)?
 A: 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.
A: 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 algebro-geometric 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 Hom-stack construction $\underline{\operatorname{Hom}}(-, BG)$.  In other words, $\operatorname{Bun}_G(X)(T) = BG(X \times T)$.
