Was Barnabas stoned wtih Paul? If not, why? Did the Jewish authorities overlook him?

I'm assuming that you are referring to Paul's stoning at Lystra (Acts 14)? If I am mistaken, please let me know.

The Bible tells us that Barnabas and Paul were together in Lystra, but only mentions Paul being stoned. The same group tried to stone "them" (perhaps including Barnabas) at Iconium before, but only stoned Paul in Lystra.

There is no mention as to why only Paul was stoned at Lystra, but I have a guess if you're interested.

Men came to Lystra from Iconium, and stirred up the people there against Paul and Barnabas. It was the people of Lystra who actually did the stoning.

The people of Lystra considered Paul to be the more outspoken of the two (they called him Hermes, because he was the chief speaker).

Paul told the people of Lystra that their religious beliefs were "vain things." That probably irritated them at least a little.

Given this, it seems likely that the men from Iconium were probably only able to find fuel to get the people of Lystra worked up into a lynch mob for Paul. Plus, given the fact that the people of Lystra believed that Barnabas was Zeus, they probably would have thought twice about throwing stones at him.

I believe this may answer your main question. If there is other information you require about Paul and Barnabas' relationship, just say the word.