Enlighten and persuade. Lawyers are trained and educated to do that. They have education, training and experience to prepare them to mediate and/or litigate.
Please note that this answer does not constitute legal advice, and should not be relied on, as each situation is fact specific, and it is not possible to evaluate a legal problem without a comprehensive consultation and review of all the facts and court pleadings filed in the case. This answer does not create an attorney-client relationship.
Even though he/she is your advocate and working for your interests, a lawyer will be able to communicate with the mediator in a more objective fashion free from the emotional component that you, as a party, carry.
Kevin King, Principal- Essential Law Services. HTTP://ESSENTIALAW.COM, 415-562-6862. The information presented here is general in nature and is not intended and should not be construed as legal advice for a particular case. This post does not create any attorney-client relationship with the author of the question answered. For specific advice about your particular situation, consult with me or another qualified attorney off-site.