What prayers can a mother make for their children to become successful in life?
Your question shows your eagerness to raise your children as Muslims who are conscious of Allah and of their responsibilities. All parents ought to work hard to do so, while constantly praying to Allah for His guidance, blessing and assistance.
Before mentioning any du'a in this regard, let me state in the outset that all parents have a responsibility to set examples for their children by providing them a home that is at once peaceful and conducive to sound Islamic training--a home where parents love Allah and His deen and do their utmost in inculcating the same in their children through precepts and practices. Ta'lim (imparting knowledge), and tarbiyah (imparting character/values)should go hand in hand in an Islamic home. The motto that children learn what they practice is an Islamic concept. That is why the Prophet, peace be upon him, taught people primarily through his own practical examples rather than through preaching. So I urge you to learn the principles of Islamic parenting and practice it earnestly and diligently, while praying to Allah for guidance and blessing. I can recommend a book for you where you can find valuable parenting tips from an Islamic perspective:
A Practical Islamic Parenting Guide - Muslim Teens - Today's Worry, Tomorrow's Hope by Ekram Bashir / Reda Bashir.
Last but not least, always pray to Allah in the manner taught by Allah and His Messenger:
Rabbanaa hab lanaa min azwaajinaa wa dhurriyyaatinaa qurrata a'yunin waij'alnaa lil muttaqeena imaaman
(Our Lord, grant us joy in our spouses and children, and make us role models for those who are conscious (of You))
Rabbi ij'alnee muqeema al-ssalaati wa min dhurriyyattee rabbanaa wa taqabbal du'aa'e
(My Lord, make me among those who establish regular Prayer and make my children among those who establish regular Prayer, and, Lord, accept my supplication)
Rabbana waj'alnaa muslimayni laka wa min dhurriyyatinaa ummatan muslimatan laka
(Our Lord, make both of us (parents) those who surrender to Your will, and make of our children a community that surrender to Your will).
I pray to Allah to bless you and grant you success in your efforts-aameen.