All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
Your sister-in-law was wrong when she prevented you from reminding your mother-in-law to say Bismillaah (In the Name of Allaah) when taking the medicine. Saying Bismillaah is permissible in every lawful matter among which is taking the medicine.
By saying Bismillaah, Allaah may bless your mother-in-law because of saying it and she will be cured from her disease, Allaah willing, and will not be in need of taking the medicine any more.
As regards saying that if she says Bismillaah before taking the medicine, she will not stop taking the medicine because of the blessing of the word Bismillaah, then this is totally incorrect and baseless.
Allaah Knows best.