Yes, it's true that the arts in question were derived from the old art of Jujutsu but I wouldn't call them closely related cousin arts. Judo and Aikido are more like 2nd or third cousins once removed while Brazillian Jujitsu is the son of Judo.
Jujutsu is a generic term used to describe the hand to hand methods that were used by the Samurai warriors. Roughly translated it means soft or pliable art. It conveys that it is the skill of achieving victory in battle by yielding. There are/were many Ryu - schools or styles of Jujutsu in Japan. Koryu - old school Jujutsu was often taught as a subart of a particular school of sword, spear, & other larger weaponry. The Jujutsu schools would teach methods of unarmed combat & use of smaller weaponry such as the knife. Jujutsu curicullum included throwing, locking, choking, striking, disarming, & ground fighting techniques.
Jigoro Kano, the founder of Judo first studied Tenjin Shin'yō-ryū Jujustsu but because of his love for Randori (grappling contests), he studied many other styles to gather techniques he felt were useful. Other schools Kano studied in depth were Yōshin-ryū Jujutsu and Kito-ryū Jujutsu. Due to Kano's love for competition, Kano redeveloped Jujutsu into a sport and method of physical fitness which he called Judo. The original Judo curricullum included Nage Waza (throwing), Kansetsu Waza (locking), Shime Waza (choking), and Ne Waza (ground fighting) with Atemi Waza (precision striking) taught to a select few in secret. Over time Judo competition began to be centered more on throwing.
Morihei Ueshiba, the founder Aikido studied Daito Ryu Aikijujutsu. Aiki means to harmonize with energy as in to blend with an opponents attack and use his energy. Daito Ryu is best described as brutal Aikido with an emphasis on stiking and hard throws. Ueshiba softened Aikijujutsu to make it into a self defense system where one could avoid harming their attacker so he could use Aikido to promote the religion Ōmoto-kyō.
Mitsuyo Maeda learned Judo from Jigoro Kano and taught Judo to Carlos Gracie. The Gracie's obviously specialized in the Ne Waza (ground fighting) of Judo. The confusing thing is why was it called Brazillian Jujitsu rather than Brazillian Judo. The following quote will offer some insight.
It is not known why Maeda chose to call his style of judo "jujutsu". One explanation is that Kodokan judo wasn't as famous in the 1920s as it is today, and that the traditional term for similar Japanese arts was jujutsu. (In Brazil, the transliteration was more often Jiu-Jitsu.) This explanation seems plausible, inasmuch as the Japanese government itself did not officially decide that the correct name for the martial art taught in the Japanese public schools should be "judo" rather than "jujutsu" until 1925.
Quote taken from.
http://en.wikipedia.org/wiki/Mitsuyo_Maeda
_Don Flatt