What should church elders/leaders do if a congregation member asks for their child to be identified as the opposite gender (or neither gender)?

Each situation is unique because each child is unique. Pastors and/or elders will want to meet with the parent(s) and listen well and humbly to them, as well as discussing with them the Bible’s view on sex and gender.

Different attitudes, different responses

It’s important to remember that the same request could be made with very different motivations. For example, a parent may hold to the Bible’s teaching but be trying to shepherd wisely a teenager who is feeling suicidal, so their request is based on a desire to enable their child to feel able to keep coming to church without it increasing their temptation to self-harm, while the parent seeks to model and teach loving biblical standards in the home.

That parent requires very different help than one who is wanting to ignore and deny God’s Word because they think that is in their child’s best interest. But whatever the situation in the home may be, pastors and elders should say they’ll be unable to comply with this parent’s request, or to ask anyone else in the church to do this, because it goes against what the Bible teaches about who this child is.

But if the situation were the first one above, this non-compliance needs to be accompanied by empathy, by prayer, and by putting structures in place to support and counsel the parent(s) and (if he/she is willing) the child. Remember that whatever the motivation of the parent, in such a meeting (or meetings—don’t assume one meeting is sufficient), your tone matters.

Stand firm upon Scripture

If the parent is opposed to the Bible’s teaching (rather than in agreement with it, but struggling to know how best to love their suffering child), and refuses to change their mind, I’d see this as an issue of church discipline, because the parent is publicly living in rejection of God’s Word. Of course, the manner and means of church discipline will vary between churches.