I am a US citizen and my mother has pending asylum case. Can I petition her?

Although you could sponsor her, you must retain counsel. Immigration is an extremely complex and technical area of law. This is not merely a matter of filling out forms. What you do not know can have grave consequences and even prevent you from obtaining immigration benefits. You need to retain an experienced immigration attorney for any immigration matter or you may throw away your future. Do not let geographic restrictions get in the way of retaining the best attorney. Pick the best attorney you can find and remember one rule: a good attorney is generally never cheap, and a cheap attorney is generally never good so don't choose based on price.