Why did the students' lawyers argue that wearing the armbands was protected by the first amendment?

Hello!

The First Amendment protects basic expression rights. Some of these include freedom of speech, freedom of religion, freedom of the press, and freedom of expression.
The lawyers likely argued that the arm bands were protected by the first amendment through freedom of expression. This freedom allows any person to express himself however he likes (through clothing, tattoos, hairstyle, etc.)

I hope this helps you!