Statute of limitations is not really the issue in this case. Typically a statute of limitations means how long after somebody commits a crime can the state wait before they file a formal charge against the person. From your question, your son has already been convicted on this offense. Depending on when he committed the offense depends if the offense is a felony or misdemeanor. Evading in a vehicle has been a felony in Texas for about 8 years.
The real question is how long does the State have to move forward on any type of arrest. When a person leaves the state of Texas to avoid apprehension any statute of limitations will also be "tolled" or stopped. The best thing you can do it contact a lawyer in Galveston or Houston area and talk to them about what needs to be done to move forward. Chances are there is an arrest warrant outstanding and at some point your son will be detained so talk to a lawyer soon.
First off, juveniles are not convicted but adjudicated delinquent. Since your son was found to have committed the offense, there is no statute of limitations nor question of arrest. If your son owed some kind of time and he fled, then he will continue to owe the time until he is caught.
You may be saying "juvenile" when that is not the applicable law. In Texas, once one turns 17, one is subject to adult laws. So, if he was 17 or older, he was not a juvenile. Again, if he was convicted, he will serve time when he is caught. And, most likely (having practiced in Harris County for 21 years), your son was given a day to turn himself in if he was convicted & then fled. This means he can also face contempt of court (for which he
can be sentenced up to 6 months) for not turning himself in as he agreed.
More often though, judges allow someone to plead, then they reset for sentencing. If that happened, then your son is subject to the entire range of punishment, as well as potentially a new case for not appearing. (It would be another felony.)
Since you specifically state "fleeing in a motor vehicle", my guess is that this is after that became a felony. (Before, we generally called fleeing from police, whether on foot or in a motor vehicle, simply fleeing.)
He needs to contact a local Galveston lawyer who can determine the actual status of the case, and go turn himself in. It will be much better for him to take care of his business by choice, than to be arrested & have to be returned to Texas (and it will cost him much less, too. He will be punished more because it costs thousands of dollars to extradite someone from another state if he is arrested in another state.)