Not to be confused with the Gruesomes -- the creepy but harmless neighbors who first appeared in the original Flintstones series in "The Gruesomes" (P-123), the Frankenstones debuted in 1979 and appeared on and off in various spin-offs and specials through the early eighties.
The first version of the family was in the episode "Fred and Barney Meet the Frankenstones" of The New Fred and Barney Show where they appeared as the managers of a condorstonium development that the boys consider moving their families into. This Frankenstone family consists of the Frankenstein-style father Frank, his wife Hidea, their giggly teen daughter Atrocia, and chip-off-the-ole-block little Freaky. Frank's voice is done in "Monster Mash"- style, and Hidea has a "Transylvanian" accent.
Another version of the Frankenstones moves next door in the special The Flinstones' New Neighbors (9/80). This time, Frank's wife is named Oblivia, and their children are Hidea and Frankenstub (who closely resemble the original Frankenstone children except that Atrocia has pale blue/white hair and seems very light on her feet.) Again, the family is odd, but there doesn't seem to be any animosity between the Frankenstones and the Flintstones, and Frank's voice is still done in "Monster Mash" - style by John Stephenson. This version of the Frankenstones appears in Fred's Final Fling in November of the same year, but -- again -- is not to be confused with the Frankenstone "monster" that appears in-between in the halloween special, The Flintstones Meet Rockula and Frankenstone.
Meanwhile, November also premieres The Flintstone Comedy Show on Saturday mornings which features a whole weekly segment starring the neighboring Frankenstones. But this version of the Frankenstones more closely resembles the version seen in The New Fred and Barney Show than the Frankenstones that just moved in next door in the prime-time specials. Firstly, Frank's wife is again named Hidea (previously the daughter's name in past specials), their two children are again kooky daughter Atrocia (now the youngest) and a now teenaged son Freaky. Plus they have a pet, Rockjaw, who devours anything. The once-soft-natured Frank is now hot-tempered and wired, with his voice now done by Charles Nelson Reilly in his traditional high-pitched hyper-whiny style. Frank now becomes volatile at the drop of a hat, especially when he has to deal with his annoying neighbor and rival Fred Flintstone. Both men are frustrated by the friendships between their wives and children -- and by the way they always seem to be thrust into one mess after another because of each other.
This series runs for two years, and consecutively with two more prime-time specials in 1981 -- which both revert back to John Stephenson's verion of Frank with his wife Oblivia, and so on. Apparently the prime-time Flintstones live in an alternate reality than the Saturday morning Flintstones -- both with their own version of Frankenstone neighbors! Just another example of bizarre Flintstones inconsistency and quirkiness.
What happens to the Frankenstones or their castle after 1982 is never explained as they don't appear in any later specials or movies. Of course, in one episode of The Flintstone Comedy Show a dinosaur picks up and runs off with the Frankenstone castle, so apparently it's mobile.