As a long time user of the free version of avast! I can definitely agree with Kees_B that the request for money is not from Avast. avast! has been protecting my computer for 8 years and Avast has never asked for money to remove anything.
The rogue program your Mom accidently installed is the culprit.
The pop up probably scared your Mom into thinking that her computer was infected and instead of ignoring the false warnings, she opened the door and let the malware in.
Free help is also available on the avast! Help Forums.
Click here to get to The Virus and Worms section of the forum.