I'm in a 12-person committee where we have to rank 5 candidates whom we have interviewed for a position. Some members of the committee have a conflict of interest, so they had to leave the room when a particular candidate was interviewed and they may not vote for that candidate (but they will rank the other candidates). What would be a fair voting system?
they may not vote for that candidate (but they will rank the other candidates)
This is a highly problematic stipulation. A voter could be accused of tactically voting for the "weakest" candidate, so that the "strongest" competitor of his/her candidate of interest has less chance of succeeding.
A system that tries to tackle that allegation, used for example in German academia, is single elimination in an exhaustive ballot (which ultimately also produces a ranked list if you exclude instant winning by overall majority). As long as the candidate of interest is still in the race, the person that declared a conflict of interest may not take part in a voting round. Only after his/her elimination, the member may return to the table.
All in all, having a committee member with a conflict of interest is always a perilous situation. Currently, there is a prominent case involving the district mayor of Berlin Mitte. For a senior position in the district administration, the mayor took part in the selection committee, despite one of the candidates being a close affiliate, and the mayor himself declaring a conflict of interest. The mayor did not issue any opinion about his friend, but otherwise took part in the proceedings. In the end, said friend was selected for the position. Promptly, a competitor sued the district, accusing the mayor to have worked against him.
The whole case came to prominence because of what happened afterwards, and which is not strictly related: The mayor tried to placate the competitor with a compensation payment in exchange for withdrawing the legal complaint. While this is a common occurrence in German labor law when private companies are involved, a public administration might not be able to act this way (at least the district lawyers ruled it out in this case, citing budgetary restrictions). Then allegedly the mayor offered to pay the compensation from his private money (he denies this, but accepts that a message could be misunderstood). He lost the political support by most parties in the district council, including his own, and was voted out of office.
I have some friends who are members of the mayor's party district organisation. Asked about the affair, what annoyed them most was that, after declaring the conflict, he did not make it his top priority to avoid any impression of nepotism. They would have expected him to withdraw completely from the committee.
From a public point of view, a formally "fair" process might not be enough. As long as the accusation of favoritism can be made, true or not, proceedings remain vulnerable.
Although there are a number of methods which are complicated by your situation here, and it's impossible for a voting system to be unambiguously fair, there are still quite a number of ranked/preferential systems which will "work" (in the sense of generating a well-ordered ranking, which might include ties).
One of the simpler options, both to explain and to implement is Score voting (also known as Range voting), providing that the scores are aggregating via averaging, rather than summation. To run this, the committee members give a numerical score for each candidate that they are eligible to judge. When processing the votes you would then take an average (typically the mean), then rank the candidates in order.
A disadvantage, and potential unfairness is that without a mark scheme, commitee members scores are likely to be arbitrary, though probably self consistent. Specifying marks in some way, even to the level of "a typical candidate should get 50, a very strong candidate 80" can mitigate this to a great extent. An advantage is that you can then trivially set a cut-off mark for acceptable candidates, useful if it's not guaranteed that the best ranked candidate will accept the position.
If time and complexity aren't a problem, then other methods can be modified appropriately, for example Copeland's method, where the committee members with a conflict don't vote on comparisons involving that candidate.
The fact that one member cannot vote for a particular candidate due to conflict of interest introduces several problems:
It incentivises the committee to enagge in tactical voting.
It incentivises the candidates to engage in strategic persuasion during interviews.
Bar the candidate in question from the get go to avoid conflict of interest.
Require the committee member in question to not participate in any voting and ask their alternate / deputy to vote in their place. This is also a way to remove conflict of interest.
Either way, the current workaround is deeply undesirable in my opinion.
Any Condorcet method should work, provided you normalize the result of each pairing in the matrix according to the number of voters that caould express a preference.
The ballot is an ordered list of all candidates from best to worst, omitting those that you have a conflict of interest with. Then you make a table where for each pair of candidates you tally which of the two is ahead in most ballots.
If some candidate "wins" all the pairings they appear in, they're the winner. If not, there are several methods to choose from to break the "loop". I'm partial to the Shulze method to resolve such situations. There are many others.
I'm not an expert on voting, but assuming you don't actually care about the final rankings but only want to have a single candidate who has the best ranking, here's one pretty solution:
The 11 jurors rank the all 5 candidates and the one juror that only ranks the 4.
First you look at the ranking from the 11 jurors, if the particular candidate is 1st then he's the "winner" if he isn't, he eliminated and is removed from the rankings from the jurors and you get the ranking out of all 12 jurors for the 4 candidates and get a "winner"