As a herald, I have been asked this question more than once. Why should I register my name and/or device? Though the question is simple, the answer is a bit more complex.
Names - Everyone has to have one
One of the most fundamental and profound things a new gentle to the Society has to decide is how they are to be known. The name is the cornerstone of their persona, which is who 'you are'. When we are born, a name is chosen for us and, like it or not, what we are known by. Some names carry with it stereotypes. However, when a gentle joins the Society, the get to choose their own identity. This is not a task to be taken lightly, for it defines the gentle, mostly in respect to others outside the group.
Why register the name? A few reasons, really. Registering the name protects it from anyone else using it. This prevents the embarrasing situation of having Robert Smyth called up in court and 5 gentles standing up, all wondering if it's them that is to be recognized. Secondly, registration of a name assures a basic minimum of authenticity. The College of Arms isn't going to register Cheryl Artemis Fitznazi de la Rocketelves of the Invisible Crystal Mountain, but if your name IS registered, there is a good chance that it COULD have existed in period. Thirdly, it prevents you from accidentally choosing a really nice name...like John the Bearkiller or Edward Plantagenet. Being called Jade of Starfall is nice, until you have to explain to His Grace why you've gone off and stolen his name. Lastly, the College of Arms needs a name to attach to the armory, and the files go by SCA name, which needs to be registered so they know where to file things.
Armory - The pretty pictures
In days of yore (which we try to recreate) a person of noble birth was known not by thier name, but rather by the arms that belonged to them. Indeed, a period heraldic treatise notes "...a knight is not known by his armor but by the coat of worship (coat of arms). which he places upon his armor." Armory is the pictoral representation of the person. It shows both who you are, what stuff belongs to you and who you are formally affiliated with.
Registering armory is as important as registering a name. It means that nobody else can wear your device or carry your badge (or anything close to it) without your approval. It also gives a sense of ownership and uniqueness to the person who has registered the device. This is MY device, it's who I am. This is MY badge, this is MY stuff. Also, like name registration, it ensures a minimum level of authenticity to the device. Additionally, it ensures that you haven't accidentally copied the badge of some hate group or errantly chosen something that a good portion of the populace would find offensive (which saves a lot of grief and fast talking).
Myths dispelled - I can't register my name/device because....
I'm not a paying member of the Society - A person's status as member/non-member only matters in one case, which occurs very rarely (once every few years, if then). If two people are registering a name or device at the same time AND they conflict AND one of the people is not a paying member, the decision will be in favor of the gentle who is a paying member.
I don't have an Award of Arms - Rank or station has no bearing on registration of names or devices. Further, any gentle, commoner or king, has the right to display their registered device. The only difference that an AoA makes from a heraldic point of view is that devices worn by people with an AoA are referred to as 'arms' and devices worn by people without an AoA are called 'devices'.
My name is already registered because it's on my blue card - What's on your card is whatever you put on your membership application. It has nothing to do with registering your name. If you put "Zwang the Destroyer" on the form, that's what they'll type on that card.
My group doesn't have a herald - Large events (Gulf Wars, Diamond Wars, Pennsic) have consultation tables with heralds there waiting to help you. In addition, you can contact a neighboring shire or barony who DOES have a herald and, if they cannot help you themselves, can get in contact with a herald who can.
My persona comes from a non-heraldic culture- Although heraldry only existed in the latter half of period, that does not mean that your early-period persona has to go without. The College of Arms registers names and devices, not personas. Also, the CoA can help you create a plausible device that will be both heraldically correct and generally fit your personas culture.
Closing thoughts
Registration of your name and device ensures that the name and device you have chosen are protected against 'imposters'. It ensures your uniqueness and affords a certain legitimacy to your claim as a noble. It lends a basic level of authenticity to your name and device as well as protecting you from making a perfectly understandable yet embarrassing mistake.