Will signing up my child for martial arts lessons give him more discipline?

Absolutely! I'm glad you posted this question...Having studied martial arts myself and having enrolled my son in the study as well. 'studying' imparts not only self-discipline but confidence and self esteem as well; and can be great in potentially dealing with "bullies". The only warning would be the demeanor of the sensei(or teacher); your child needs to 'fit' with him/her so meet with first and decide.
Maybe. As in all coached sports, the quality of the coach will determine the quality of your child's experience. Respect, personal responsibility, and discipline are emphasized in most martial arts programs; in fact they are also emphasized in most other sport programs. Help your child select activities which are fun for him or her. Your child may need to try different ones out to find a 'best fit'.