Does the title of "Marquis" change the name of its holder?
Sort of, yes. The naming system for British peers is quite complicated and confusing to those who haven't encountered it. Every person has their own name and surname, but when they inherit, or are given, a sufficiently high title, then the name of that title is generally (at least in formal communication) used for the person instead of their own personal name.
A historical example to illustrate the point: Arthur Wellesley was awarded a peerage (firstly as a marquess, later as a duke) for his military achievements. He (or more exactly his brother acting on his behalf) chose to name his peerage after Wellington, a town in Somerset, as the name was reasonably close to his family surname and there was a manor available there. Subsequently he was referred to as "Lord Wellington" or "the Duke of Wellington" rather than as "Sir Arthur Wellesley". The peerage title of Wellington replaced the family name of Wellesley, when addressing or referring to this man.
A Marquess (sometimes spelled Marquis in the French style) is the second highest peerage rank after Duke. "Marne" is a fictional Marquessate, not among the real ones in Britain and Ireland, but the general form of address for a marquess would be as "Lord" followed by the name of their title, so in this case "Lord Marne". The title holder would have an actual name, like "James Mair", and Mair would be the family surname, but he would be generally referred to as "Lord Marne" after receiving the title. Cf. the current Marquess of Winchester, whose name is Nigel Paulet but who would be formally addressed as Lord Winchester. (Sometimes, just to make it even more confusing, the family surname may match the peerage title.)
is "Marne" the name of their family, and whoever remains alive from the family becomes "The Marquis of Marne"?
Marquessates are usually (always?) hereditary, and certainly the fictional one in this story is since James inherited it from Maurice. Therefore, yes, the title would continue to be passed on through the family until/unless the family line died out. (For example, the title of Marquess of Willingdon, the most recently created Marquessate, was established in 1936 for Freeman Freeman-Thomas, then passed to his only surviving son, who never had children, so the title was abolished when he died in 1979.)
But other members of the family would still be called Mair, unless they also received noble titles. Some peers have multiple titles, in which case they use the highest-ranking one, while their sons may be entitled to use some of the others. (Recall that Duke > Marquess > Earl > Viscount > Baron.) For example, the current Marquess of Salisbury also has a right to the title of Viscount Cranborne. He uses the Marquess title since a Marquess is higher than a Viscount, but while his father was alive he was referred to as Lord Cranborne (not having inherited the title of Lord Salisbury yet), or even Baron Cecil (the lowest of the associated titles) when he joined the House of Lords, and his elder son and heir currently uses the title of Viscount Cranborne. So if James Mair had a son, he wouldn't necessarily be just "Mr Mair" until receiving the title of "Lord Marne"; he might have some other subsidiary titles. But certainly Marne is not the family name; it's a title held only by one person at any time.