I believe that I gathered from what you posted is:
- There are three viable candidates as far as leadership ability.
- Your traditional model is for the "retiring" team captain and lead mentor to choose.
- A significant number of team members do not like two of the candidates.
- The current team captain does not feel that the candidate who is liked by the team is the best choice.
- You have not said anything about the lead mentor's opinion on the matter. As you are asking the question, I would infer that the lead mentor does not have a strong opinion, and also that you are the lead mentor, or someone whose advice is usually considered by the lead mentor.
First of all, let me preface this with a statement that there are probably no two teams that do things exactly the same way, so YMMV.
Unless you already have a history of a team vote for captain and are happy with it, this is about the last selection method I would use. Democracy is a messy enough system for deliberative societies - for teams it's usually unworkable.
Given that the outgoing captain has made a selection, the question becomes one of whether that selection is the right person to lead the team. The first thing to know is
why this candidate is unpopular. Unless these reasons are incompatible with this candidate being an effective captain, I'd ratify the decision and move forward. I can think of a number of attributes of a good captain, but which ones are more important depend greatly on the team dynamics. Here are a few top reasons we use on our team, which follows a more military than bureaucratic model:
- Is the candidate a good example (shows up consistently, shows GP, works hard, does community service, deals with people and machines and frustration well)?
- Is the candidate respected by the team members? (Respected, not liked)
- Is the candidate able to delegate specific authority to lieutenants and team members for appropriate tasks, then follow up at an appropriate interval?
- Does the candidate handle pressure well?
- Does the candidate have enough force of personality to stand up to any of the other team members when (s)he's right, and good sense to yield gracefully when (s)he's wrong?