I've talked at some length about the use of Who..Whom in Jury Nullification before, which I'd like to expand on more.
Specifically, how does one actually carry this out?
Unlike FIJA, which is a fine organization, I'm not advocating openly attempting to nullify. Instead, I'm advocating a backhanded and passive aggressive nullification strategy, one that is far harder to fight against, by, say, the judge removing you from the jury. Conflict requires both in your face and slippery strategies---one enables the other just like rushing sets up passing opportunities in American football.
The first rule is you never breathe a word about jury nullification anywhere near the courtroom and certainly not in the jury deliberations. You don't even hint at it with things like---oh I don't think it's right to send him to jail for so long for X or I'd rather use the prison space for guys like Y. No, instead what you do is expand the area of fuzziness that you are given into obscene proportions. That fuzzy area is 'reasonable doubt'. What does reasonable doubt mean? Does it mean being 99% sure? 99.9%? as sure as a Christian with 4 aces?
The key is it means precisely what you want it to mean. So take a leaf from Public health guys making 'studies' about gun control. Start with your desired conclusion and set your parameters to whatever they need to be to 'justify' it. Remember, if you torture the data enough, it'll confess to whatever you want it to. Of course you go who...whom, under the same circumstances as the FIJA-style nullifier, you just aren't open about such.
The first phase from your perspective (I'm assuming you're not a grand juror---I've been called for jury duty several times but never for a grand jury) is jury selection. Both the prosecutor and the defense are looking to shop for the most favorable jury they can get. What you want to do is betray as little information to them as you can short of obvious perjury. They're looking for information out of the verbal band as well as what you actually say. For instance, the more words you say, the better their estimate of how smart you are is going to be. If they're actually good at their craft (fortunately, an AWFUL lot of them are horrible at it, I can normally infer a person's political positions and cultural alignment within a minute or so of hearing them speak on matters that are not explicitly political, it is a matter of word choice and inflection and a willingness to combine signals that are only in the range of 80% or so accurate into a reasonably accurate prediction, stereotypes are damnably useful things) the more you say the more likely your game is up. Plus particularly smart people are prone to be more verbose and to want to cover their answer completely and with the desired nuance. Try not to present that image. Last time I was in jury selection I knew one of the sides was going to smoke a peremptory challenge on me when the defense attorney asked me what standards of evidence I was familiar with employing, and I answered him with the two that he knew (preponderance of evidence and 'reasonable doubt') and another that he didn't know (clear and convincing, typically an administrative law standard midway between the other two). But I had negative interest in being on that jury. So remember, short answers that betray no bias towards defense or prosecution, even though if you're going to go Who...Whom, you are the most biased defense juror imaginable.
Assuming you make it onto the jury, what you need to start doing as the trial progresses is make lots of notes (if they allow you, many don't), or mental notes (if you have to), on anything and everything that is plausibly painted as weak in the prosecution's case. Imagine what the five star defense attorneys would question. Question that and remember it for the jury deliberation. It's actually better if the defense doesn't actually question it, because then the prosecutor would have a chance to defend against that line of argument. But guess what, neither lawyer gets to go anywhere near the jury room. You do. Think 'motivated skeptic'. Emulate whatever group you consider to be 'immune to reason' that 'uses its high intelligence to deny what is nearly certainly true'.
Here's the bottom line. Simply by staying the course, you can almost certainly force a mistrial, since the jury won't reach a decision. But you can play for more than a mere tactical victory. You've got things on your side, notably that you're likely way more committed (imagine this is, say, a case against some redneck for having a shotgun that is 1/4 of an inch too short) than are the other jurors. They know this (and you should NOT explicitly remind them of this, that is likely to backfire)---they get to leave only once a mistrial is declared or a decision is made unanimously. One way to get there is to surrender to your who...whom and return a not guilty verdict. In addition, by spinning scenarios that are admittedly of low probability, you will make them seem more real to the other jurors, and hence higher probability in their minds. Drip, drip, drip--exploit those cognitive biases. Try to seem reasonable and extremely 'fair minded'---it helps if you betray a lot of cultural markers of disliking the class/group from which the defendant is drawn, this will make your scenarios wherein the defendant isn't actually guilty seem more credible, as in an admission against interest. But if you can't swing the other 11, no biggie, a mistrial is usually nearly as good as an acquittal. Done in reasonable numbers, this procedure would result in making nearly any law without an overwhelming consensus behind it practically unenforceable. The only defense against it is trying to take away the right of a trial by jury, which, by the way, was one of the big grievances motivating the 1st American revolution.