Let's back up. The person whose name is on the title is the legal owner of the vehicle. The legal owner of the vehicle in most states [and under common sense] has the duty to insure the vehicle in the name of the owner of the vehicle. If other people will be operating the vehicle, the insurance company needs to be notified and the names of the additional drivers must be included on the face of the auto insurance policy as authorized drivers.
(Sometimes people play games by "putting" another name on a car and saying someone else is the "real" owner. I caution you to follow the above reasoning. Insurance companies can and do disclaim coverage if an unauthorized user is operating that vehicle. For example, in one case a 17 year old was driving the parents' car and was in an accident. The parents knew putting the 17 year old would increase their rates. When the 17 year old was in an accident the insurance company disclaimed coverage, costing the parents more in the long run.)
This answer is offered for informational purposes only. It is not offered as, and does not constitute, legal advice. Laws vary widely from state to state. You should rely only on the advice given to you during a personal consultation by a local attorney who is thoroughly familiar with state laws and the area of practice in which your concern lies.
I would like to reiterate what Mr. Myers said. If you put the title to the car in your name and get the insurance in your name and your boyfriend's son drives it all the time, the insurance company may refuse to provide insurance coverage. You need to tall the insurance company that your boyfriend's son will be driving the car and have him added on the insurance policy as an additional insured.
Now to your question. First, anybody can be sued for anything. Your question is would you be liable for damages caused by your boyfriend's son driving a car you own. In Oregon, there is a doctrine called the "family purpose doctrine". Very generally, under this doctrine parents are liable when their children are in accidents while driving the parents' car if the children were living in the parents' house at the time. If you are not related to your boyfriend's son that doctrine should not make you liable.
However, there are other theories under which you might get sued including negligent entrustment or allowing your boyfriend's son to drive your car when you know they should not. E.g., if they are drunk. Another theory might be negligent maintenance or allowing your boyfriend's son to drive your car when the brakes don't work or there is something else mechanically wrong with the vehicle that makes it dangerous because you did not maintain it.
The bottom line is. If you put your name on the title to the vehicle, you are responsible for that vehicle and may be subject to liability if something goes wrong with it. I suspect your boyfriend does not want to put the car in his son's name because he wants to try to save money on insurance. As Mr. Myers noted, that's a bad idea.
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