I am a naturalized U.S. citizen, and my daughter, who is 17, still lives in our native country. She has a one-year-old child who was born out of wedlock. If I sponsor my daughter for a U.S. green card, can she bring her young daughter in as well?
Unfortunately, the answer is "no" (but read to the end of this article for alternative strategies). This is because of a technical (and often illogical) aspect of immigration law, to do with which immigrating relatives can bring their own "derivatives" (spouse and minor children) to the United States with them on the same application.
In cases of immediate relative relationships (which include the spouse, parents, and unmarried, minor children of U.S. citizens), derivatives are never allowed. The U.S. citizen must be able to directly file a visa petition (Form I-130) for every person that he or she wishes to help immigrate. In order to do that, however, the citizen's relationship with the "beneficiary" of the petition must be recognized as a qualifying one under U.S. immigration law. And the law simply does not recognize grandchildren of U.S. citizens as having a qualifying relationship for I-130 filing and green-card eligibility purposes.
As we mentioned above, however, there are strategies that might help your daughter avoid separation from her child, though it means her path to a green card will take longer.
The first is for your daughter to wait until she is 21 or over for her to immigrate. In that case she will not be an "immediate relative," but instead be classified in the "family first preference" (F1) visa category. Preference relatives are legally allowed to bring their derivatives (spouse and ummarried minor children) with them to the United States. The down side to this strategy is that there is an annual limit on F1 visas, as a result of which a waiting list has developed, and it will likely take several years after you file the I-130 for her to be able to move forward with a green card application. You'd need a lawyer's help to work out the details of timing this correctly.
Another, similar possibility is to wait until your daughter gets married, in which case she would be classified under the "family third preference" (F3) visa category. Because this, too, is not an "immediate relative" category, she would be able to bring her spouse and child. But the waiting list in this category tends to be even longer than in F1.
These are complex strategies, so again, your best bet would truly be to consult with an immigration attorney for a full analysis and help in properly preparing and filing the paperwork, and working out how best to time the whole process.