How can a person deal with his mother forcing him and his siblings to lie to relatives, claiming envy, while he desires to please Allah and avoid causing the family to break apart or severing his relationship with his mother?

Lying is forbidden, and persistence in it leads to the corruption of both religious and worldly affairs. It is not permissible to engage in it except in specific circumstances. Therefore, it is not permissible to obey one's mother if she orders lying, because there is no obedience to a creature in disobedience to the Creator. The impermissibility of lying must be explained to the mother gently and respectfully. If one fears harm from speaking the truth, equivocation (tawriya) and insinuation (tarid) can be used instead of outright lying.