Voevodsky's original definition of the algebraic $K$-theory spectrum, $KGL$, was given as follows:
The component spaces were fibrant replacements of the infinite Grassmannian $BGL$. The structure maps were then defined by using the projective bundle theorem for algebraic $K$-theory and the fact that you can lift maps between cofibrant-fibrant objects in the homotopy category to honest maps (use the injective model structure so that all objects are cofibrant).
I'd like to understand what happens to the spectrum KGL as the choice of lift of a map in the homotopy category varies. The initial issue is that a map of spectra is defined by diagrams which commute on the nose, not just up to homotopy.
As a side remark, it's known now that there are other models for $KGL$ (Bott inverted projective space), so what I really care about are the comparisons one can make between prespectra which have the same component spaces and homotopic structure maps.