What does the adenosine do in ATP? This is an extremely basic question about biochemistry from someone who doesn't know much about organic chemistry. I'm aware of how organisms store energy ATP and release it by dephosphorylating it into ADP. My question is about the role that the adenosine part of the ATP molecule plays in this process.

Naively, it seems as if the triphosphate group could be attached to any old thing, and you'd get the same energy by dephosphorylating it. But as far as I'm aware, it's only ATP that's used for this purpose in biology, and not any other triphosphate. Why is this? Is the fact that the triphosphate is attached to an adenosine just a "frozen accident" like the genetic code, or does it play an important role in the phosphorylation/dephosphorylation reactions themselves?
The adenosine part is needed to make DNA and RNA.  It is logical for the triphospate to be attached to ADP, GDP, UDP or CDP since they are needed to make DNA and RNA.  

In some instances, GTP is used instead of ATP.  In fact the Krebs cycle does involve phosphorylation of GDP.   GTP is used in the formation of microtubes by tubulin.

I've seen some give the rate of hydrolysis being greater for ATP than GTP as a reason for ATP being more utilized.  

There is a related question here: https://biology.stackexchange.com/questions/11286/why-is-atp-the-preferred-choice-for-energy-carriers