Do you have any conference rooms with visibility in from the outside (“fishbowls”)?

"altering behavior solely because a colleague is a woman goes against such ideals" Then you should have no qualms here, because this is about her personal convictions, which she holds independently of her gender.

Good answers here. Regardless of what you choose, document it. You'll be treating this person differently because of her gender and religion, that obviously looks bad out of context. Prefixing an email with "as per your request that we do not meet alone" should suffice, but check with HR.

@Emilio I dont want to make her gender and religion into a big issue. My goal is to seek balance and not focus on how she is different from other team members

Your priority is, as it should, seeking balance and productivity, keeping peace and harmony in your team. But don't completely disregard covering your legal rear if things go completely sideways. Imagine what it would look like in a lawsuit if she can prove that you did 1-on-1's with everyone but her, and you can't prove that she asked for it. There's no need to mention her gender and religion, nor mention her motivation, but do write down that you're accommodating a request of hers.

I don't want to give a protracted answer, but it feels to me you are conflating company values with personal beliefs to some extent. From my reading of the situation it is entirely possible to keep in the spirit of the company's values, and at the same time completely respect her personal ideals without placing yours above hers. Think of it this way: some of what you are proposing may sound as disrespectful to her, as much as what she is proposing may sound disrespectful to you.

So are you suggesting you being able to speak freely and being free from distraction (and her privacy and her being able to speak freely, which I imagine she'd more than capable of considering herself in her decisions) is important enough to justify going against her ideals and making her uncomfortable? Otherwise it seems trivial to just allow a 1-on-1 in a more open space.

@NotThatGuy to a certain extent. I want her to feel she can talk to me privately without fear of intrusion from coworkers and I am someone she does not need to be especially deferential to, just because I am male. I want to help her integrate into team without drawing unnecessary attention to her religion or discomfort with private interactions with folks of a different gender

@Haukinger She is not a "trouble maker" at all. In Islam, an unmarried woman is not allowed to associate freely with men in private. We used to have a female Muslim whose father (or another family member) brought her to and from work every day, because public transport was off-limits. FWIW she was better engineer than most of the "white males" in the department.

One key way to respect someone is to let them decide what's best for them themselves. If she feels comfortable speaking openly in a less private space or she feels her ideals are simply more important, that should be her decision to make. Meeting with her in a less private space shouldn't draw much, if any, attention or discomfort if you don't make a big deal of it and you let anyone else do the same if they want to.

@Emilio, I did make a note. Exactly, by not showing special preference by KEEP having 1:1 , not isolating her by focusing on differences

Don't focus on equality to the detriment of inclusion. Equality AND inclusion, not either or. If this means making an honestly small accommodation for her, what's the issue?

I have a sense that even some non-religious women (e.g., in the U.S.) might ask for this same accommodation.

Your ideals are a bit too rigid. It's like saying "I support marriage equality! I want any man to be allowed to marry any woman!". Well, yes, but some people have no interest in the configuration that you chose. Accommodating different needs is part of what is commonly understood as equality.

You might consider that not everyone is all that good at "open conversation", and for your team member, the male/female religious thing is only going to add to that. Have you considered just holding the interview via email &c, where neither of you will be distracted, and don't have to respond in real time?

I wonder if a more public area that still allows some privacy would be okay? I am thinking about seatings in the foyer, in a corner in the break room or in a cafe next to the office.

I find it interesting that this is a sticking point for something as small as a 1:1 meeting, I wonder what will happen when a situation arises where this person must be alone with another male employee/customer to meet the requirements of the business? Certainly seems like something you would want to involve HR with going forward.

I've thought about this a bit. Private meetings in public spaces are not a solution, as they imply a relationship too. The best solution to honor her request would be to invite her husband to the 1 on 1, but that is highly impractical and could be seen as not a honest attempt to honor her wishes. In any case, involve HR now. She's already asked for religious accommodation whether one sees it differently, HR must document it and document what was done for accommodation.

We had a lady do about the same thing in one of my former employer's offices. Basically it came down to not having the meeting at all. Later she underperformed on her tasks, and when being admonished for her performance, she decided it was sexual harassment because a man was her (unhappy with her) manager. This roped in a different team mate as a person under suspicion because he used the word "girl" The accusation came nine months after the supposed word was uttered. It was a mess.

Basically, nobody could recall the exact details of the meeting except the lady being processed for underperformance. HR did their best to keep everyone as happy as possible, and the company lost lots of time and money in the process. The accused coworker was very distraught (he was an introvert, and barely interacted with other women being a super-devout catholic who felt it was a violation of his bond with his wife). The team dissolved and the accused left for a new job two months later.

@Anthony You don't want to make her gender or her religion an issue; but, in short, she's made both of those the issue. At this point you are just along for the ride. Don't add wood to that fire, but bring it up with HR now. Being directed by your organization into solutions is a requirement for the organization to protect itself in case the well meant solutions spiral out into unforeseen paths.

Would video recording the meeting count as having another person present?

@Anthony I would highly recommend checking out Islam stack exchange