The reason some lawyers don't like mediators is that so much of the marketing for mediation services "piles on" the widespread anti-lawyer, anti-court system rhetoric that has been out there a very, very long time (even before Shakespeare went for the cheap laugh in Henry VI). Actually, mediation works quite nicely within the litigation system and is a valuable tool in the lawyer's belt.
Similarly, the often-expressed frustration of mediators is that lawyers are disruptive to the process -- that they are too adversarial when cooperation would make our jobs easier.
But the truth is, the attorney is not supposed to make our jobs easier. The attorney is one of the mediator's clients in the mediation process. As mediators, it is our job to make their jobs easier.
Functionally, the lawyer's role is to assist their client in analyzing the value of their case and to help them obtain that value through resolution of the conflict. Practically, the mediation is often the client's only "day in court" -- and the client expects, wants, and gains confidence by witnessing the lawyer performing that core function -- advancing the client's cause, standing up for the client's interests, and helping secure an appropriate value through mediation.
Attorneys are also important in helping educate their clients about the best and worst case scenarios absent a negotiated settlement -- what mediator speak calls the "Best/Worst Alternative to a Negotiated Agreement". Lawyers can help clients clearly understand the options and various scenarios out there, in part by offering a slightly less emotionally-involved analysis of the facts, law, and fact-finder's likely range of outcomes. They thus provide the foundation and the framework for the client's settlement analysis, helping to give the client some clarity of goal in the sometimes confusing, often frustrating process of negotiation.
Mediators who negatively view the participation of attorneys in the process are missing out on an important opportunity to advance the negotiation. Very successful mediators understand that, if they provide the opportunity for lawyers to discuss the case analysis and to advocate on behalf of their clients, the lawyer's participation can become part of the foundation for the negotiations and very useful in moving the discussions towards resolution.