In general, you should dress very conservatively, even if it is not what you would wear on a daily basis. This is not dishonest, it is respectful. You would dress more formally to go to synagogue or a job interview, and this meeting is basically a combination of the two. Further, your dress, even if it is not "where you're at" right now, it shows that you know where the end goal is. And don't worry, they will ask you about your daily clothing choices, and you should be honest.
A separate point we've briefly mentioned before is relevant here: another consideration (a controversial one) is that the beit din will also be evaluating your "marriageability." That means try to look your best, rather than throwing on your muumuu or other frumpy clothing. I have a hunch this is true even if you are converting "with" a partner, unless you and that partner are already married. Marriageability isn't that significant of a factor, but it's part of the larger consideration of whether you will "fit in" in the community and be happy socially. Not finding a marriage partner is one of the major causes that will push an otherwise-successful convert off the derech. So it's an important issue to consider (and warn the conversion candidate about), though it is hard to define and the concept angers many people.
What does "conservative" translate to? Let's discuss it separately for each gender. Depending on the feel you get from your beit din, you may dress more casually for future meetings.
Men: You'll have to forgive me, I'm not as "up" on men's clothing. Commenters are especially welcome to make additions to this. You don't necessarily have to wear a suit, but you should if you are planning to convert chareidi/yeshivish/chassidic. If you're going to wear a suit, go for black. Gray can work, but why bother when you almost certainly own a black suit? Stick with the simple, and try to wear neutral colors. Avoid bright red, even ties. If you want to dress more casually (and even in the most liberal of orthodox conversions, you should still consider wearing a suit since you would wear one to a job interview), dress business casual. You should at least wear dress pants and a collared shirt. Again, stick to neutral colors. You should not wear shorts or distracting clothing. Your shirt should be buttoned up or otherwise come to your collarbone.
Women: You have more leeway to wear what would be considered more "casual." You also have an easier time in that there are firm rules to follow, which can decrease your anxiety about what to wear. (But if these rules are stricter than your community's interpretation of tznius, you may be annoyed. Too bad, get over it. Remember, what works in the club or hair salon doesn't work for a job interview either.) Your clothing should cover your elbows (not just reach the elbows), cover your knees even when sitting, and cover your collarbone. You should be wearing a skirt. An even more conservative idea to consider is to avoid a top that mimics cleavage, even if you're wearing a shirt under it. In other words, avoiding v-neck tops, even though you are fully covered. Some communities hold even the suggestion of cleavage is un-tznius. That's not true in most communities, but again, you need to dress very conservatively for these meetings. Most conversion candidates will be fine with such shirts so long as your are properly clothed otherwise. You should not wear bright red, but you may want to avoid other day-glo colors as well. Aim for neutrals, even though you do not need to wear blue, black, and white for most batei din. If you want to convert in a community that does believe tznius requires wearing only black, navy blue, and white/cream, then you should reflect that in your meetings. But for most people, that will not be the case. You should wear closed-toe shoes that also cover your heel. That is also stricter than many communities' standards, but you should be wearing the same in a job interview.
Headcoverings: Men should wear a kippah or other community-appropriate clothing. A baseball hat and many other hats are probably too casual to be your headcovering for this interview. Married women should cover their hair, following the standard of the community if possible. If you don't own a sheitel (wig) and that is the community standard, a hat should still be fine. But expect that other types of headcoverings may enter the conversation.
Jewelry: Men who wear jewelry should wear it in traditionally-male ways and wear very conservative jewelry, if at all. You should likely remove ear piercings or other piercings. Necklaces should not be too noticeable. Cufflinks should be simple and tasteful. Beyond that, I can't think of male jewelry. Female jewelry should also be simple and tasteful. Unusual piercings should be removed. It is alright to wear "Jewish" jewelry such as a star of David.
In general, you should dress in a way that would fully acceptable in your intended community. This will show that you understand what the community standard is and that you are prepared to live within it. This is one of the subtlest and most powerful ways you can show your research and what you know about the community. Just like in a job interview, your clothing can set the stage or serve as a distraction. Ideally, what you're wearing will seem so natural that they don't even register it. Don't let your clothing be distracting or a black mark against you. This is a place where it is important to remember how to pick your battles. Your beit din is not the appropriate group to fight with if you disagree with how your community (or others) interpret tznius. They did not set that standard; it is just their responsibility to ensure that you fit in to the community you claim you want to join.