Can I apply for a visa for my mother who was deported in 2005?

If you mean to apply for an immigrant visa (in other words, for permission for your mother's coming to live in the U.S. permanently) you can file the petition unless the deportation was for an aggravated felony or for a serious immigration violation (such as illegal re-entry after removal, fake marriage, etc.) There is no need to file separately for her minor child so long as the child is included in the petition and supporting documents. From experience, when a case involves prior deportation, it is not a good idea to undertake it as a do-it-yourself project. At the very least, have an immigration attorney check your papers before filing them.