Can I sponsor my son who is 10 years old if I am only a green card holder?

Yes.

It will take a few years before he would be eligible for an immigrant visa.

Yes, you can. However, depending on your home country, it could take anywhere from 7 to 20 years before a visa is available. I strongly suggest that you contact an experienced immigration attorney for a face-to-face consultation and give him/her all of the facts surrounding your situation. He/she would then be in a better position to analyze your case, advise you of your options and, if possible, make sure that the necessary paperwork and documentation is presented so as not to delay the process.

Yes, you can file the I-130 for your son as a green card holder. It is simply a lot slower because you must wait for the visa to be ready (Something around 2.5 years right now) between the initial filing of the petition and the time to get an interview for the immigrant visa.

As an LPR or green card holder, you can sponsor your son by filing an I-130 petition for him to come to the US. Such petition is in the 2d family preference F2A category and current processing times in that category for most countries is about 25 months. After the petition is approved you may then start consular processing at which time someone may have to accompany him to the US embassy in his country for issuance of a visa. Given his tender years, the US embassy in some countries have certain procedures in dealing with such cases and so, you'd need to be familiar with those policies as may be applicable to the US embassy in his country. If you become a US citizen at any time while his I-130 is pending, his visa category changes automatically as he becomes the child of a US citizen thereby eliminating additional wait times. You should always consult with a lawyer for more specific advice tailored to your individual situation

More information is needed to answer your question well. You may be able to file an immigrant visa petition on his behalf, but, depending upon when and how you came to have the greencard, there may be other options.

You can sponsor your son in the F2A category. Your son, however, will have to wait in line to receive a visa. Check the Department of State's visa bulletin for February to see how long he may have to wait in line.

Permanent residents can petition their children, however the process is not as quick as when a US citizen petitions his/her child.