There are many facets to this question. There are hundreds of articles on the internet and in other publications which list important considerations like experience, competency, subject matter expertise, style, track record, cost, etc. These are all very important factors. No doubt.
I come at this question a little differently because many of the cases I work on are multi-faceted and involve an extensive cast of parties with divergent interests. In these cases, I think it’s important to step back and carefully consider what is impeding settlement. Is there a fundamental disagreement on valuation, or perhaps it has proven unusually difficult to communicate effectively with other parties, or else maybe there are client relationships interfering with the negotiations? It’s much easier for a third party mediator to objectively assess the dynamics of the dispute.