Is it okay to go solo in research?
There are two components to this question. "Is it okay to work on my own 'independent' project in a PhD?"

Yes, definitely. The purpose of a PhD is to train an independent researcher. An advisor who feeds their student the tasks or jobs to do throughout their PhD is probably doing them a disservice, as they would not have the opportunity to learn how to devise and carry out their own independent research projects. Being in the third year, you have probably finished your classes and accrued enough research experience to do so. Of course it is expected that you keep your advisor in the loop if you do decide to work on such a side project.

Now, with regards to "What would my advisor think if I worked on my own 'independent' project?"

That depends entirely on the advisor and your funding situation. If you are full supported by a specific grant, it is probably a bad idea to work on an unrelated topic. Ideally, you should work the problem into something which can be related to the grant. Of course, if you are self-funded or have a no strings attached fellowship, things can be more flexible for you.