What is the ruling on vowing to fast Mondays and Thursdays if the one making the vow is unsure whether he uttered the phrase "every Monday and Thursday," and is it permissible to offer expiation for it?

If the vow is conditional upon your mother not becoming angry, then it is a vow of justification intended to draw closer to Allah in gratitude for a blessing received. Since you do not remember whether you said "every Monday and Thursday" or not, what is doubtful is considered non-existent, and the default is freedom from obligation. So, if you fast on Mondays and Thursdays once, you are absolved of your vow, unless you become certain of the contrary.