Detailed answer to your question can be found on pages 59-60 of City of Splendors: Waterdeep. While it is 3e sourcebook, it might be reasonable to assume that how one becomes a noble would not change much in Waterdeep, given its long history starting with the nobility claims of various wealthy families dating back to 882DR and the formal beginnings of the nobility dating back to 1248DR, when the Lords of Waterdeep recognized the merchant gentry.
Nobility was established as a practical way of diminishing chaotic use of self-proclaimed titles and as a carrot to lure powerful land-owners from the surrounding countryside and wealthy merchants from far away lands into the city. As these goals got more or less achieved, the rate at which the Lords ennobled new families has slowed down, though it is still possible for a wealthy (min 25,000 gp personal wealth obtained from mercantile endeavours) individual to ask the Lords the title of nobility for themselves and their descendants. There are no written rules for this, but based on examples, some rough criteria can be deduced as described on page 60 of the book.
Perhaps the most important of these criteria is being sponsored by at least five noble families, and current families do not want to reduce the privilege of being nobility by adding new families. So ascending by petition is quite unlikely in modern Waterdeep and that leaves marriage as the only practical way for an ordinary citizen to achieve nobility.
Each family has a head noble, a patriarch or matriarch. Whether and how a relative of this person gets to be considered a part of the noble family is unclear, presumably this is decided by the head. Some families have ~10 nobles, others have over 50. Titles, lands, and funds can be inherited by any child or heir of the head. The standard practice is the eldest child (regardless of gender) inheriting the title and the majority holdings.
Available 5e Information
It is worth noting that none of the data in this answer are contradicted with the information on pages 55-57 & 154 of the 5e Sword Coast Adventurer's Guide. While we learn that during Lord Neverember's tenure as the Open Lord, it became legal for impoverished houses to sell their titles, this decision was reversed with Laeral Silverhand and it is still difficult to become nobility. Quoting:
Although they seldom agree on much, one matter that all the noble houses see the same way is that their status should not be tainted by newcomers, and certainly not by anyone so brightcoin as to purchase one's way to a noble title.
Further evidence that the century between CoS:W and SCAG has not changed the nobility much comes from SCAG's explicit mention of "More than seventy-five noble families call Waterdeep home" (note 75 and not 80 for instance), and the still-freely-available web-enhancement for CoS:W lists exactly seventy-five noble families. (This web enhancement is rather nice for a DM who wants to develop the historical background. It lists the founding year, ethnicity, trade and interests, predominant alignments and favored deities, and resource limits of each noble family as they were in 1370s.)
PS: SCAG describes the Waterdhavian Noble background on page 154, where we again read that "... there are around seventy-five lineages in Waterdeep". And one needs to be in the main line of a family to be eligible for becoming its leader one day.
