From March 31, Meghan Markle and Prince Harry will no longer be senior members of the royal family. Though they'll keep their Sussex titles, they're losing their HRH titles—which means that, for the first time in his life, Harry will take on an official last name. Though he went by "Harry Wales" at school, HRH title holders are not legally required to have a last name (although modern royals often choose to use Mountbatten-Windsor, a combination of the Queen's and Prince Philip's names). As for Meghan, she's still referred to using both her maiden last name and her Duchess of Sussex title, and she could continue to use both.
The most obvious choice for the family is to collectively adopt the last name Mountbatten-Windsor, which has been used since the '70s. The Queen is, of course, a member of the royal House of Windsor, and Prince Phillip's naturalized last name is Mountbatten. Per the royal family's website: "For the most part, members of the Royal Family who are entitled to the style and dignity of HRH Prince or Princess do not need a surname, but if at any time any of them do need a surname (such as upon marriage), that surname is Mountbatten-Windsor."
That said, both William and Harry went by "Wales" at school. That's because Charles' title is Prince of Wales, and the stand-in last name for royal family members is often the title of their parent. That tradition carried on to William's family: At school, Prince George is known as "George Cambridge," because his father's title is the Duke of Cambridge. So, even though their last name is technically Mountbatten-Windsor, their informal last name is Cambridge.
But the plot thickens! When Harry and Meghan's son, Archie, was born, they did not give him royal titles—perhaps in a nod to events to come; perhaps because they wanted him to have a more normal life—which meant that the baby needed a surname. Enter Mountbatten-Windsor, which is formally and informally Archie Harrison's last name. It sounds very formal, but it's actually a rebellious move—if Harry had followed in William's footsteps and stuck rigidly to royal tradition, Archie would have taken on his title (the Earl of Dumbarton) and been known as Archie Dumbarton.
Fast-forward to now, when Harry, Meghan, and Archie have essentially opted out of being senior royals and therefore need last names. Archie is already set—he's Mountbatten-Windsor, in a nod to his great-grandparents. But then there's Meghan, who is still known by her maiden name, Markle—although since she married Harry, she's stopped signing her last name "Markle"—and who also goes by the Duchess of Sussex (a title she'll maintain), and therefore doesn't have one set last name. She might choose to go by the last name Mountbatten-Windsor, Sussex, Markle, or something completely different (which would be a break from tradition, but when you've already walked away from an entire institution, what's the big deal?).
Either way, Meghan "Markle" differs from the last names of both Harry and Archie right now, and she may want to keep the same last name as her family. Harry doesn't actually have a clear last name right now—again, it's technically Mountbatten-Windsor, but he's used "Wales" when he's had to use a last name throughout his life. He's going to have to determine once and for all what his official last name may be, which may spur Meghan to take the same one. And that name could be anything—Sussex and Mountbatten-Windsor are the most likely options (and the latter, of course, would be a link to Archie), but they could also take Markle or a completely different name of their choice. (This is Harry and Meghan, after all.)
Along with being Sussexes, the couple are also the Earl and Countess of Dumbarton, and The Baron and Baroness Kilkeel, so let's throw Dumbarton and Kilkeel in the mix as potential last names. They may have considered a break from senior royal lineage through giving their new family a new last name, but Archie's last name is already Mountbatten-Windsor, so...