In John 8:58 - 8:59 it says "Jesus said to them, "Truly, truly, I say to you, before Abraham was born, I am. - Therefore they picked up stones to throw at Him, but Jesus hid Himself and went out of the temple."

Originally, according to the Gospel of John they did in fact try to stone him for blasphemy but failed as he went into hiding out of the temple.

More however, according to several Gospels (specificly Mark) - Jesus was trialed by the Roman Governor due to the fact that the Pharisees were not allowed to execute or commit an execution or such during Passover as it would go against one of their laws which were in place and so they went to the Roman Governor and let him have the formal execution as the Romans were not Jews and saw no problem doing an execution during Passover.

Sources: UMKC.EDU

Since blasphemy was a crime punishable by stoning in Biblical times

Claiming to be the Messiah--God's chosen leader of the Jews--wasn't blasphemy. Many modern Jews believe that such a person will some day arise to lead them.

Why did they let him rise to such power and favor

Did he? The grandest demonstration of his power was a minor disturbance at the temple.

why did they bring him to formal trial with the roman governor?

It is not clear that the Sanhedrin actually had the power to execute people at the time, due to the Roman occupation. And in the event that they did, it is not clear that Jesus committed any of their capital offenses, such as blasphemy. The Roman authorities, on the other hand, were always eager to publicly execute rebels.

Jesus had followers who believed he was the Messiah, and had caused a public disturbance at a politically sensitive time and place. It is easy to see why the local authorities would rather hand him over to the Imperial authorities before his movement caught the attention of their attention on it's own and brought the military might of Rome down upon Judea.

The simple answer is that Jesus most likely never claimed divine status, at least not publicly, and the religious leaders of their day were not authorized to execute male prisoners without consent of the Roman Authorities.

Long answer:

First, we should discuss the consensus of what current historians believe the historical Jesus said and did. To do so, I am going to pull out a brief description of him from a research paper I wrote a short while back.

Consequently, historians generally agree on only a few basic and subtle pieces of biographical information on the historical Jesus. Among them, is a belief that the historical Jesus was a minor religious figure and teacher who primarily taught around rural parts of Galilee, however he likely made pilgrimages to Jerusalem. Jesus of Nazareth was an apocalyptic prophet, who's primary message preached that the end of the world was near, and he preached his message to his closest friends. After upsetting the local Jewish leaders like so many other Rabbi's had up until this period, he was found guilty of crimes against the state when we was tried before the Roman governor of Judea and was executed by crucifixion. Historians estimate that at the time of his death, he had very few followers possibly between 20-50 people, and was probably not very well-received by the people in his immediate area until the years after his death.

This excerpt also addresses one of your assumptions in your question which is "Why did they let him rise to such power and favor." The truth is, they did not. At the time of his death, it's estimated that Jesus only had about 20-30 followers. It was in the decades after his death that word of his accolades spread throughout the eastern part of the Roman Empire which eventually led to about a 5% conversion rate of Rome's population by the time Constantine Converted around 313 CE. So in essence, I am saying that they did not wait for him to become popular -- they killed him as soon as they saw a threat. And it is also believed that the actual crime that was reported to them by the historical Judas was that Jesus was preaching that he would one day be king over Israel, not that he was God, so punishing him for blasphemy would not be appropriate. You can also see this since the only reoccurring statement between the dialogue between Jesus and Pilate in all four gospels is that Pilate asks, "Are you the king of the Jews?" Now Jesus' answer is different in each gospel, but what we learn here is that in all likelihood, Jesus' crime, as reported to the Roman governor was plotting to overthrow the Roman Government and become King. It's obvious that the Gospel writers wanted people to believe that Pilate did not think this was plausible, hence why Pilate doesn't want to execute Jesus, but still, it is universally accepted that from this, we can learn his true supposed crime. (A more detailed answer of this can be found in Bart Ehrman's book: Jesus, Interrupted: Revealing the Hidden Contradictions in the Bible (And Why We Don't Know About Them)

So to answer your second question, during the feast of Passover each year, the Roman Governor (in this case Pilate) would travel to Jerusalem from his headquarters in Caesarea with armed soldiers. This was done because as emotional stories were told over the freeing of God's chosen people from the Egyptians were remembered this time of year, insurrections were common. So to get a strong hold over the area, Pilate would come to Jerusalem and govern from there during this time each each year. So the issue with Jesus, in this particular case was, if he was being accused of a crime against the state (being king), it would go before Pilate since he was governing from the city during the Passover. However, it was Roman law during this time that any man being accused of any crime against the state would be tried before Roman authorities since Jewish leaders were not permitted to do so.

edits: spelling