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As adorable as the name Archie may sound to some -- and as much as Meghan and Harry clearly like it -- it does break with some precedents.
Firstly, no member of the British Royal Family has ever been given a first name that most people consider to be a nickname. "Archie" is usually short for Archibald. Secondly, even the name Archibald is almost without precedent. You have to stretch back five centuries to find it in the family tree. The Scots lord Archibald Douglas 6th Earl of Angus married Henry VIII's older sister, Margaret Tudor, after her first husband, King James IV of Scotland died. This made Archibald the stepfather of King James V. He and Margaret had a daughter, also named Margaret, who married Matthew Stewart 4th Earl of Lennox. Their son Henry Stuart Lord Darnley famously (infamously?) married his cousin Mary Queen of Scots and fathered King James VI of Scotland, who later succeeded Queen Elizabeth I of England as King James I of England. The two thrones were later merged. The new little Archie is descended from this original "royal" Archibald through several lines through both his grandfather Prince Charles and his late grandmother Lady Diana Spencer.
Thirdly, the name Harrison is even more clearly without precedent. There is no great-great-great grandpa Harrison in the royal family tree. However, the name may just be a nod to the age-old tradition of recognizing a child by its father's name. Marvel fans may be familiar with Thor Son of Odin. Or, perhaps you have heard of Leif Ericsson? Ericsson was not his surname or his family name; it literally meant that his father was named Eric. The name Harrison literally means "Son of Harry" and you couldn't get more spot-on than that. Many cultures share this patronymic (father's name) tradition: Abu or Ibn in Arabic, Ben in Hebrew, Ap in Welsh, -vich in Slavic languages, etc. Fitz meant "son of" in Norman French and was used as a part of surnames for some royal bastards as recently as Queen Victoria's first cousins, the plentiful FitzClarences, children of King William IC who were born when he was still the Duke of Clarence. Even the prefix of Mc-, Mac- or O' so common in the British Isles originally indicated who the father was. The use of formal surnames that passed from generation to generation tends to be a more modern concept, especially for non-noble classes. At the time surnames were adopted, people who had a patronym, just passed their own down to their children, freezing the that genealogical marker in time. (Other surnames were commonly derived from location names or jobs but even these mark a certain point in history. For instance, I have known many people named Cooper, but none of them made barrels for a living.)
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Finally, little Archie Harrison received only two names. While that's quite common for most of us. It is not as common for babies with royal daddies. The Queen's children and Prince Charles's children (including Harry, who is named Henry Albert Charles David) each have four names. Prince William's, Prince Andrew's, Prince Edward's and Princess Anne's children each have three. The Queen has three names (Elizabeth Alexandra Mary), but her younger sister, Margaret Rose, had only two. Princess Anne's oldest grandchild as three, but her other granddaughters only have two each. Queen Victoria only had two (Alexandrina Victoria) but her cousin/granddaughter-in-law, Queen Mary, had eight (Victoria Mary Augusta Louise Olga Pauline Claudine Agnes).
Of course, the baby's announced surname, Mountbatten-Windsor, also raised questions among some who are less familiar with the history of the British Royal Family. Both names were "made up" in a sense in 1917, when the Queen's grandfather, King George V, decided that the members of the British Royal Family needed to drop all of their Germanic names and titles in the midst of World War I. The royal family itself did not really have a surname -- royals didn't need them -- so they really had to make something up. Ultimately, they decided to lay claim to the most historically British thing in their midst and named themselves after their home at Windsor Castle, which had originally been constructed by William the Conqueror shortly after his Norman Conquest in 1066. As for the royal cousins who had been born Princes of Battenberg, they merely translated their German name to English. Since "berg" means "mountain", they renamed their family Mountbatten. One of their princesses, however, was already married to a Greek prince. During the next World War, her son, Prince Philip of Greece fought in the British Navy. Afterward, in the run-up to his marriage to the heir to the British throne, he surrendered his foreign citizenship and his royal house. He opted to adopt his mother's family's newish surname of Mountbatten. When his engagement to Princess Elizabeth was announced, he was simply Lt. Philip Mountbatten. (His father-in-law made him Duke of Edinburgh and an HRH and made it so that the children of the marriage would be HRH Prince/Princess.)
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Everything seemed fine for a time. Four and a half years after the wedding, Princess Elizabeth became Queen Elizabeth and some people wondered what her surname was as married women traditionally took their husband's name. When Philip's uncle Louis Earl Mountbatten of Burma boastfully toasted the Royal House of Mountbatten, the actual Royal Household and senior royals were aghast. Parliament quickly decided that Elizabeth and her children were all of the Royal House of Windsor. Philip is said to have felt emasculated. "I'm just a bloody amoeba!" he declared at one point in response to this and other slights. He was, he noted, the only man who could not give his name to his children. This is thought to have been a rough point in the royal marriage for a long time, until shortly before their third child, Prince Andrew, was born in 1960. Elizabeth issued an Order in Council declaring that their male-line descendants who lacked royal titles and styles would use the hyphenated surname of Mountbatten-Windsor. Since then, the name has appeared from time to time on wedding registeries and birth records of many family members who DO have royal titles. It is the name of Prince Edward's children who do not use their royal styles, however, his daughter is usually just referred to as The Lady Louise Windsor. This means that little Archie may be the first descendant who may actually use the combined surname throughout his entire life.
The bigger issue, in my option, is not the baby's name but his title. Or rather, his lack of title. Although many people wished that he would have been HRH Prince Archie of Sussex, like his Cambridge cousins, he actually wasn't entitle to a royal title. Royal styles currently only extend to the children and male-line grandchildren of the monarch and to the children of the Prince of Wales's firstborn son. In order for the child of the Prince of Wales's second-born son to become a prince, new Letters Patent would have been needed.
However, calling Meghan and Harry's baby just plain-old "Master Archie Mountbatten-Windsor" flies in the face of centuries of aristocratic tradition. The eldest son of a duke, royal or otherwise, is traditionally allowed to use his father's next highest secondary title. In this case, Harry's next title is Earl of Dumbarton. The daughter of a duke is traditionally entitled as Lady First Name Surname and younger sons as Lord First Name Surname. (The children of the Queen's cousin the Duke of Kent are the Earl of St. Andrews, Lady Helen Taylor and Lord Nicholas Windsor. Even Prince Michael of Kent, way down at #48 in the Line of Succession these days, has children who are style Lord Frederick Windsor and Lady Gabriella Windsor.) If Meghan and Harry wished for their children to have no titles at all, they could have refused to accept the titles that Harry was granted just last year. Of course, under current practices, this would have made Meghan's title Princess Harry. I can only imagine the uproar that would have caused on at least two fronts: one that she was 'forced' to use her husband's name (as Princess Michael of Kent does) and two that she 'outranks' her sister-in-law, the future Queen Consort, who is merely Duchess of Cambridge. Both of these are fallacies of sorts. Until extremely recently, all married women used their husband's name (Miss Jane Taylor married and became Mrs. John Smith). More importantly, and I want you all to remember this, both Catherine and Meghan ARE princesses because their husband's are princes. Period. They are styled as Duchesses, rather than as Princess William and Princess Harry.
Over all of this title business is looming another issue: the future. As it stands, no matter what Archie is called now, he is his father's heir. Unless he predeceases Harry or surrenders his rights himself, he will inherit the Duke of Sussex title at some point in the hopefully distant future. Much sooner than that, Archie will stop being the great-grandson of the monarch and become the grandson of the monarch. As you'll remember, male-line grandsons are entitle to be HRH Prince. What will happen the Prince of Wales becomes King? Will Master Archie suddenly become Prince Archie, or will he be the only member of the immediate royal family without a title of any kind?