It serves as a some kind of "handle" for enzymes.
I mean, there is a common conserved domain among enzymes using adenosine cofactors (ATP, FAD, NAD etc.) called "nucleotide - binding fold". It binds to adenine or adenosine, respectively. So basically, adenosine part serves as a not-so-specific handle of the cofactor and the active part (nicotinamide, for example) ensures the specifity part of reaction. The reason for this is that nature can design lot of different proteins with specifity to different cofactor just using this single protein fold.