If I remember correctly, to be referred to as a pure-blood, all your grandparents must be magical. If you have one (or more) pure-blood grandparents and one (or more) Muggle grandparents, then you are a half-blood.
What I am wondering is this; two Muggle-borns marry and have children. Those children then have two magical parents, but four Muggle grandparents. What are these children considered? Muggle-born or half-blood?
There doesn’t seem to be any specific name that the pure-bloods use for the children of Mudbloods. When Yaxley, a Death Eater, refers to them, he calls them “brats of Mudbloods”.
“Mother to Maisie, Ellie and Alfred Cattermole?’ Mrs Cattermole sobbed harder than ever.
‘They’re frightened, they think I might not come home –’
‘Spare us,’ spat Yaxley. ‘The brats of Mudbloods do not stir our sympathies.”
- Harry Potter and the Deathly Hallows, Chapter 13 (The Muggle-born Registration Commission)
This implies that there’s no specific name for them, otherwise it’s likely Yaxley would use that name if it existed.
As far as we know there is no special term for this type of children. Or their children. They will likely be called "Mudbloods" or "Half-Bloods" by Death Eaters unless they want their allegiance for some reason and "unpure" and the like by others with a slightly racist mind. Most likely everyday wizards will simply not consider them an honourful wizard family with a long family history like others, and just consider them "normal" unspecial wizards.
For the Half-Blood categorisation see also What is the blood status of a person born to a half-blood and pure-blood?
In Chapter Seven of Chamber of Secrets we have the following statement:
“It’s about the most insulting thing he could think of,” gasped Ron, coming back up. “Mudblood’s a really foul name for someone who is Muggle-born — you know, non-magic parents.
Given that Ron defines a Mudblood as someone with non-magic parents, the implication would be that non-magic grandparents do not make someone a Mudblood.
On the other hand, the underlying idea of blood-status is that some people diluted their pure wizard blood with Muggle blood. If your blood is merely diluted then you’re a Half-Blood, but if you’re blood is entirely Muggle then you’re a Mudblood. If all four of your grandparents are Muggles then your parents likely had no wizard blood. If that’s the case then (particularly in the last book where Muggle-Borns are thought to have stolen magic from wizards) the fact that your parents are wizards wouldn’t purify your blood. It would stand to reason, then, that you’d still be a Mudblood.
Of course, neither of these arguments are definitive.
Rowling has stated that she based the Death Eaters’ hang-ups about blood purity based on grandparents directly on Nazi propaganda about Jews. For example, the Death Eaters’ concept of Half-Bloods corresponds to Mischlinge in the Nuremburg laws.
If the analogy is between Muggles and Jews, the closest real world analogue to the child of two Muggle-borns would be the child of ethnic Jews whose families converted to Christianity. In Nazi (and Death-Eater) propaganda, this makes no difference. Those people are still racially inferior. That ideology made a point of not having a different term for them suggesting they were different—if anything, they were seen as presenting a special danger of marrying into the Master Race and making it impure.
If you are instead looking for a neutral, socially-acceptable term, I would suggest second-generation.
Well, in the movies there are only three terms mentioned:
Mudblood for a mixed child of wizard and Muggle (like Hermione)
Pure-blooded for children of wizards (Weasley)
Squib for born in the wizard but not a wizard (Filch)
But there is no such description given in movies for Muggle parents (Dursley)
I would choose new blood. If they continued having children with muggleborns then wouldn't it be the start of a new magical family. For example take lastnames of the muggleborns to be Miller and Martinez. If continued to marry other muggleborns they would have started the new blood magical family 'Miller' or 'Martinez'. This is just my theory and my opinion.
