No significance. All my yarmulkes are from family weddings and bar/bat mitzvahs. I’ve just kept the ones that were nice.
No significance whatsoever, except for personal significance. The same goes for the very wide variety of tallitot (the prayer shawls).
For example, it’s not uncommon to see Gator-themed yarmulkes here in Gainesville.
My daughter just had hers a few months ago. I’m not Jewish, but my wife is, so there you go.
Yeah, the big thing is that now you’re grown up and can read from the Torah for the first time. But the reading you’re supposed to do is determined by the week of the ceremony, so it’s going to be some random thing. The kid has to read in Hebrew, with a watcher ready to smack them if they make a mistake. And then it’s customary to give a speech discussing the particular Torah portion.
My daughter’s portion? All about the correct way to rape captive women and when it’s OK to stone your son to death.
For bnai mitzvot (bar and bat mitzvahs) and weddings they will often have ones stamped with the name and date as a keepsake.
Completely insignificant. People’s own yarmulkes are often decorative and ornamental, with fancy embroidery. The ones you find in the box by the door are more likely just plain black crepe-paper.
At my home, as a child, we lit Sabbath candles and said a few basic prayers every Friday evening. My elderly grandmother, from the Old Country, was there sometimes, so she had the honor of saying some of the prayers. She covered her head with a paper napkin.
Do go to the service–both of you The bat/bar mitzvah boy/girl has studied and practiced for a couple of years to learn exactly what to do and how to lead the service and read from the Torah scroll (which is in Hebrew, of course). It’s a huge, ginormous accomplishment. The whole family will be glowing with pride. Don’t miss it. Expect to get a little teary-eyed and possibly choked up.
Aside: Bar (son)/ Bat (daughter) Mitzvah (commandment) = “Son or daughter of the commandment”
Mitzvah is sometimes used to mean “good dead,” but it actually means “commandment.” Plural is “mitzvot” but the plural of Bar/Bat Mitzvah is B’nai Mitzvah- “B’nai” is children [sons + daughters]. It’s like the plural of Attorney General is Attorneys General.
Of further passing interest: The word “bar”, meaning “son”, isn’t even Hebrew. It’s Aramaic. (The Hebrew for “son” is “ben”.) I’m not sure if “bat” (for daughter) is Hebrew or Aramaic or both.
There’s actually quite a bit of Aramaic that’s crept into Jewish tradition. I went to an Orthodox Jewish wedding once. The wedding “contract” was written in, what looked like Hebrew to me. Tradition requires that there be some witnesses, and my brother was one. He said the document was actually Aramaic. (I’m pretty sure he would know that.) Some parts of the Old Testament were originally written in Aramaic.
There’s no reason a parent of an invitee can’t attend the service. At a conservative or orthodox ceremony, it would be considered good manners for a male to wear the yarmelke in the box. If the box is just inside the entrance to the building, it’s conservative/orthodox. If it’s just at the entrance of the chapel, then it’s probably reform. The Oneg afterwards is open to everyone who attends the service. There may be a reception line where attendees can congratulate the Bar/Bat Mitzvah and his/her parents. This would be a good opportunity to slip the guest of honor an envelope containing a card and check. Don’t bother trying to find a Bar/Bat Mitzvah card at a Hobby Lobby. A Hallmark store would be better. There’s not much selection though, be forewarned.
As for the party, it could involve a sit down meal with all sorts of relatives and teen guests and a dance party or just a teen dance party with some light refreshments. It’s been longer than I care to remember since I did the Bar Mitzvah circuit and the rules may have changed. Back then, it was coats and ties (or Bar Mitzvah suits) for the boys and party dresses for the girls. Mini skirts were the style then. 
Much more recently, my nephew wore a tux for his BM party.
The best advice I can really give you is to recommend you give the Bat Mitzvah girl’s mother a call and explain to her you’re unfamiliar with what’s expected and ask what questions you have. You might want to write them down first.
For the record, the ceremony isn’t really required but it’s become a tradition. A child becomes an adult at age 13 (12 for girls) which means they’re now expected to follow the rules, customs, and traditions. Look, as long as you complete the coursework and get the necessary grades, you’re a high school graduate whether you attend the graduation ceremony or if they mail you the diploma. Now, for Sabbath services, you don’t really need a Rabbi to lead the services. It’s traditional to invite new members of the congregation or a guest to the Bemah (altar) to take a brief part of the service. The BM child is considered to be a new member.
My understanding is that Aramaic is to Hebrew what Tex-Mex is to Castilian Spanish. Or Cockney to the Queen’s English. IOW, basically the same language. Hebrew was only spoken by the ancients in Temple ceremonies and rituals (that would be THE Temple), and Aramaic was the lingua franca of regular people on the street. But essentially the same language. Like you might use street slang in your daily life, but presumably you would speak standard English at a White House dinner (even if you wore flip-flops).
Can someone back this up, or straighten me out?
If you take one its yours to keep. Usually each person having a life event like a Bar/Bat Mitzvah or wedding, gives a supply as a customary gift (usually with their names and occasion dates imprinted on the inside).
Thereby providing every Jewish family with a dozen nonmatching kippahs in their junk drawer, half of which memorialize the weddings of people they’ve never met.
Aramaic is a closely related language, but it is not the same language and I don’t think they are mutually comprehensible. A better analogy might be Spanish and Portuguese.
A friend of mine who speaks Hebrew fluently told me that in a supermarket (in New Zealand, actually) he heard some people speaking a language that was clearly close to Hebrew but wasn’t really understandable. He finally asked them (in English) what was the language and they said it was Aramaic. It turns out that there is a significant Aramaic speaking population in southern Iraq. Of course, modern Aramaic may not be that close to the biblical language. One Jewish prayer that I know is in Aramaic is the mourning prayer.
As for the OP, above all remember that this is an occasion of great joy and no one there will be jeering at any misstep. I have actually been to only one bat mitzvah. They didn’t do them 65 years ago when I had my bar mitzvah. That one was a new Reconstructionist congregation (basically conservative except for equality of women) that was held in a Friends meeting house. It was followed by a fairly quiet reception back at the girl’s house. No DJ, no dancing.
If there is a basket of yarmulkes by the door, it is polite to pick one up and wear that. In a Reform synagogue there will be none.
Thanks for clearing that up, Hari Seldon. Todah Rabah. 
My daughter got how to get rid of household mold and how to cure venereal diseases!
The procedures are similar.
For more information on the history of Aramaic and Hebrew, here’s a couple YouTube videos I like. Though notice it’s by an enthusiast, not an expert–he’s in college learning this stuff.
Is this universal? I’m not Jewish, but I work Jewish events, and I’ve seen baskets of kippah/yarmulke at Reform synagogues as well as Conservative ones (I seem to hear the Hebrew word “kippah” more around here.) The difference I’ve noticed around here is that the Conservative congregations I’ve been to required me to wear one (and I believe there was a note near the basket saying as such.) With the Reform, I’d ask and I was just told it wasn’t necessary.
It’s not universal. The reform synagogue my family used to attend always had a box of kippot available just outside the sanctuary.
I believe the difference is, as Reform Judaism generally puts more of weight on personal autonomy in terms of religious practice, most reform congregations leave it as a decision for the individual worshipper. I recall seeing men not wearing them, and even some women who elected to wear them during services.
I can confidently state that this is not universally true. Plenty of reform synagogues have the baskets.
At my 150-year old Reform synagogue, kippot (pl. of kippah) are available in a basket by the door for those who want them. A few men wear them. I’m guessing that 30+ years ago you definitely would NOT find them there. When the Reform movement started in Germany ~1825, the idea was to look less “Jewish”-- no Hebrew, no special garments, organ music allowed, etc. Ultimately, that didn’t work out so well. In recent years, Reform synagogues have been going back to their roots-- lots of Hebrew, kippot, tallitot, the whole shebang.
Thanks again everybody! Everything went well and my son had a good time.
The service was nice, and shorter than I expected after hearing of 3 hour services (only an hour). The Bat mitzvah girl was nervous but did great in my admittedly clueless opinion. She sure had a lot of memorized Hebrew to recite!
The party was held in a large teen rec area of a Jewish Community Center, so I was able to find myself an unobtrusive spot in the back to keep a low-key eye on my son. The mom was very nice and understanding, and told me to help myself and get something to eat.
It was good to see my son get to interact with his peers in a social situation and proud of him that he did well and didn’t do anything to spoil the party. I was nervous because one of the guests was a kid at his school who does not get along with my son, and they have had some pretty bad incidents at school. But they were both on their best behavior.
Mazel tov! I’m glad your son had a good time and that you both got to experience something new.