Since she is not working and does not have an employer to apply to extend her work permit, she would only have the option to apply to change her status to visitor. She should apply for this a month before her work permit expires and include the information that she is in her last trimester of pregnancy and can not travel. Processing her application will likely take around 2 months so by the time they get back to her on that, she may have had the baby already.
The only problem with this is the question if she will keep her health care coverage if she changes her status to visitor. There was another woman on this forum who was a temporary worker with an expiring work permit and was also pregnant. She had talked to EI, Service Canada and the health care in her province and was advised to change her status to visitor and told that she would still keep her EI maternity benefits as well as health care but it may depend on the province. This woman should ask health care about this where she lives. If she ends up without coverage, it could end up being very expensive for them to have the baby in Canada, I have heard $10,000 or so.
Having the baby in Canada will not get her citizenship. Changing her status to visitor is a temporary solution. Eventually she will get sent home unless she finds another job and gets a new work permit or she marries your cousin and he sponsors her for PR.
If she goes back to the Philippines to have the baby there, your cousin can still register the baby for Canadian citizenship. I do not know what rights he would have in the Philippines. He will have to work it out with the mother. If they end up on bad terms, he may have to go to court there to get visitation.