What does the "b'dekin" at a Jewish wedding accomplish? Does it play anow Halachic role in the marriage? Or is it just some sort of ceremony that's meany to symbolize something?
Remember what happened to Yaakov when he thought that he was going to marry Rachel and wound up marrying Leah. She was veiled and he did not realize that it was Leah until after he had consummated the marriage. At the Bedekin he himself puts the veil on her face. That action shows this and ensures that he is indeed marrying the woman that he expects to be marrying.
Note that the Hebrew word בדוק translates as check. This may one reason for the name bedekin. I have not researched the etymology so this is just a comment from what I have noticed.
There are also a number of symbolic meaning to the bedekin
The Veiling Ceremony (Bedeken) Rabbi Maurice Lamm goes into detail
According to several rabbinic authorities, the veiling was not a mere social formality, but had the force of law as it was legally considered the chuppah, which is the concluding step of the marriage. Most authorities today hold that it is a beautiful traditional ceremony, but that it does not have the significance of chuppah. Nonetheless, its performance assures that all theories of chuppah are observed, to guarantee the absolute observance of the law of marriage, without exception.
There are a number of interpretations of the veil's symbolism, all of which reflect truths that are worthy of being dramatically enacted before the wedding service.
The veil is a symbol of the married woman.
Rebecca does not wear a veil while on the journey in the company of the servant, Eliezer, but instinctively dons it when sighting Isaac. This may account for the insistence of major authorities that the groom himself veil the bride, and that it should never be done without him—it is only his presence that makes her veil significant
The veil is symbolic of her new unapproachability to others, not only sexually, but as hekdesh, a sanctified object in the temple.
The symbol of the veil most often referred to is "modesty."
The veil also conveys psychological significance. Netziv notes that the instinctive action of veiling at the sight of Isaac symbolized Rebecca's married life with him.
The Rashash in Kiddushin (18b) says that the minhag of Badekin is rooted in the kinyan - acquisition - of Prisas Taliso Aleha - Laying the garment upon her.
The Bach (61) explains that it has the halachic status of Chupah.
Although this is not agreed upon by all authorities, nonetheless, Rav Asher Weiss explains that we do it to cover all opinions.
The Drisha (65) refers to this practice as well:
נמצא מה שהולכין בזמנינו כל הקהל עם החתן או עם אביו בשחרית לכסות ראש הכלה היא היא הנקרא בזמנינו הכנסת כלה לחופה ע"ד שכתב המרדכי שמשחרית מתחיל הכנסת כלה לחופה שע"י זה אנו מקדימין לומר חופה קודם לקידושין ואע"פ שכתב רש"י ללוותה מבית אביה דמשמע שהולכין אחריה אפשר שכן היה המנהג בזמנם כמו שנוהגין עדיין בקהילות בארץ אשכנ"ז שיש להן בית מיוחד לזה שמביאין אותם שמה:
The Dagel Mervava (Yoreh Deah 342) says that the custom in Russia was for the Groom to cover the Bride with garment, and that perhaps it effects the marriage as chuppa.
The Avuharam (ברכות אירוסין ונישואין) also connects the minhag to Prisas Taliso Aleha
Rav Shmuel Vosner (שו"ת שבט הלוי ח"ט סי' ע"ר) even goes so far as to say that one should set aside eidim - witnesses, to witness the groom place the cloth over her head, as it, acc. to many halachic authorities, has the significance of Chuppa.
So in short yes, according to some, it is significant and can serve a Halachic purpose.