We don't know with certainty what it would have meant (the commandment itself being our only source from that period), but there are several likely theories on what it meant. What I consider to be the most likely is the one mentioned below by TheApiary, that it is saying to not swear an oath or covenant falsely under God. Common practice for these in the near east around that time period was for a person make an oath or agreement to another with a deity or sometimes another person as a guarantor that they will carry out what they swear or be punished by that guarantor. The commandment then almost becomes a matter of keep your word, a sentiment that may be echoed in Jesus' later statement in Matthew 5 (the sermon on the mount, which is in many ways concerned with interpretation of the commandments) verse 33

“Again, you have heard that it was said to the people long ago, ‘Do not break your oath, but fulfill to the Lord the vows you have made.’ 34But I tell you, do not swear an oath at all: either by heaven, for it is God’s throne; 35or by the earth, for it is his footstool; or by Jerusalem, for it is the city of the Great King. 36And do not swear by your head, for you cannot make even one hair white or black. 37All you need to say is simply ‘Yes’ or ‘No’; anything beyond this comes from the evil one.

The latter half of TheApiary's statement is also true, the Israelites (and most modern Bible translations) would not say the true name of God (the tetragrammaton, which is where we get Jehovah and Yahweh from today) and would instead say Lord (Adonai), which is also likely taken from an interpretation of that commandment (in my Christian thinking, incorrectly, like many Jewish law interpretations).

Another possibility is that it meant to not take on an identity as one under God (an Israelite) when you were not sincere about it. Do not take on or swear an oath by the name of God if he isn't your only true God so to speak. This would be to prevent the name of God being brought into disrepute by those who aren't truly his follwers and keep it highly regarded and holy (a sentiment common in all interpretations, even the first I mentioned).

The last one, and honestly the one I find most suspect, is that it was to keep God's name from being used in magic. The names of the gods were frequently used for magic protection spells (such as written on amulets), for all manner of magic rituals, prophesying, any sort of practice that needed a supernatural power. This commandment was to stop God's name being used in those, and by extension, since He was the supreme supernatural being in Hebrew thought, stop his people from performing such rituals (which have other laws against them in the biblical books of law and are against the way of thinking and reverence for God and His plan that He requires of his people). It was probably a legitimate concern in this period (and even later ones, the Kabbalah uses the name of God for magical rituals), but I don't think it is enough of a primary concern to be included with the other commandments, which are all really big, society affecting ideas and concerns (another thing which leads me to the oaths interpretation).

Just a quick final note, I don't think you should be so quick to say ancient Israel wasn't replete with teenage girls saying "oh my God". One of the most common mistakes people make in history (and anthropology) is that they think there is too much difference between people and their thinking in different time periods and cultures. Teenage girls in ancient Israel would not have been all that different to teenage girls today. They probably wouldn't have been using anything like "oh my God", for various reasons I won't get into, but I just wanted to put this point out there.

You may want to try r/academicbiblical as well.