There really are only two direct verses/passages that talk about a mother staying at home with her children. Titus 2:3-5 says, "Likewise, teach the older women to be reverent in the way they live, not to be slanderers or addicted to much wine, but to teach what is good. Then they can train the younger women to love their husbands and children, to be self-controlled and pure, to be busy at home, to be kind, and to be subject to their husbands, so that no one will malign the word of God." The other direct verse is 1 Timothy 5:14, which says, "So I counsel younger widows to marry, to have children, to manage their homes and to give the enemy no opportunity for slander." Another translation for the phrase "to be busy at home" in the Titus 2 passage is "to be keepers at home."
Also consider some less direct verses. Proverbs 14:1 states that it is wise for a woman to invest in her home. We see the priority that God places on the home and the woman's involvement. Clearly the home is not to be neglected for the sake of outside employment. Deuteronomy 6:4-9 teaches the importance of constantly teaching our children. Of course, this is addressed to fathers as well as mothers. Staying at home with your children would only give you more opportunities to teach your children God's ways. So it is a positive investment that you can make in your children's lives by literally applying this passage of Scripture.
It certainly is a commendable thing for a mother to be at home with the children and devote herself to training them full-time. Women are definitely encouraged in Titus 2 and 1 Timothy 5 to stay at home with their children. A woman should keep her home a priority and not let it fall by the wayside.