How can an 18-year-old brother who is negligent in prayer be helped to commit to it, especially since he resorts to excuses and delays some of the obligatory prayers?

You have done well by caring for your brother's improvement, encouraging him to pray, and supplicating for his guidance. Add to that:

1. Inform him that abandoning prayer is an act of disbelief (kufr) that exposes its perpetrator to the Fire.

2. Bring him books and tapes that clarify the ruling on one who abandons prayer.

3. Encourage him to keep company with the righteous.

4. Connect with one of the righteous so that he may link your brother with righteous people or beneficial lessons.

5. Show him your keenness for his guidance and his happiness in this world and the Hereafter.