Can I petition my mother if she came here illegally?

You need to consult an immigration attorney to discuss the specific details of your mother's case. Based on the information you have provided, it is impossible to say what, if any, options your mother has.

I have found that there is almost always an avenue to correct errors of the past. And it may be true in this instance as well. Your mother's long attachment to this country and presence of qualifying US citizen relatives including yourself are all positive factors that will be considered including the possibility that you or other immediate family relatives can file a Form I-130 petition for her. But, also note that DHS will consider the bad things she might have been done. You did not elaborate on the nature of her felony and you can be certain, that the government will take her criminal past into consideration. This is to be expected. In my opinion, your best option is to consult with an experienced immigration lawyer with your mother who must give all accurate information regarding her criminal and immigration history. Doing nothing is not helpful because, given both her illegal status and criminal history, it is just a matter of time before she comes to the attention of DHS

If you are a U.S. Citizen you may be able to get her a green card but she may need a waiver. Approval of a waiver will depend on your family situation and any evidence of hardship. You should consult with an Immigration Attorney who can walk you through the process and help you prepare the necessary legal paper work. There is a new waiver procedure that was released in March of this year which makes applying for the waiver easier (she can apply here in the U.S. instead of leaving the U.S. and applying for the waiver at the consulate).