I'm not sure if this is what you are after, but when I started to look at Grothendieck-topologies I thought of being an open immersion as a topological property; somehow it should be possible to recover all open immersions from the topology, and if we changed the topology to the etalé site, the same method should give us the etalé maps as "open immersions".
Unfortunately, I doubt this is possible (it would be kind of nice if it were, so please correct me if I'm wrong). The reason why I was fooled to try, was probably due to the topological flavour of the term open immersion. But passing to the Grothendieck topology loses information about our model covering maps we started with. For instance, if {U_i -> U} is a covering, then a sheaf F would satisfy the sheaf condition also for the set {U_i -> U} U {A -> U}, the latter being an arbitrary morphism.
Instead I think it is more correct to think of the property of being an open immersion as something that we know what it is in our base category (affine schemes) and want to generalise to our new, larger category.
That said, we must look for the correct definition of open immersion in our base category. We want it to be something that is the "complement" of a closed immersion. Let F be a subfunctor of spec A. Pulling it back along spec A/I -> spec A should give us the zero scheme. Now define the complement of spec A/I as the most general subfunctor F of spec A satisfying this (i.e take the categorical limit). I've not done the details, but I suspect this gives the right concept and that a concrete description of what the subfunctor looks like would be as in exercise VI.6 in E-H. Extending it to representable morphisms of sheaves (in any reasonable topology) is now straight forward using pull-backs.
