Neither medieval manuscript calls this collection anything. The word mabinogi appears as a closing formula at the end of each of the Four Branches, and its meaning is itself uncertain, possibly "a tale of youth", possibly matter concerning the god-name Maponos, possibly the material a trainee storyteller learned. At the end of the first branch, in one manuscript, the form appears as mabynnogyon, treating an already-plural word as a singular and pluralising it again. That single occurrence is generally agreed to be a copying mistake.
The mistake became the title. The form was used in print in 1795 by William Owen Pughe, and Lady Charlotte Guest adopted it for her translation of all eleven tales, published in parts from 1838 to 1845 and collected in 1849. Guest was an English ironmaster's wife who learned Welsh, worked with Welsh scholars including Tegid and Thomas Price, and produced a version that carried the tales into English and thence into European Romanticism and, eventually, into a great deal of modern fantasy writing. Her translation is bowdlerised in places and her Welsh has been superseded, but the name she used has proved permanent, and it now denotes a grouping no medieval Welshman made.
The manuscripts themselves are later than the tales. Linguistic evidence puts the Four Branches in the eleventh or twelfth century in something like their present form, and Culhwch and Olwen earlier still. Some fragments survive in Peniarth manuscripts of the mid thirteenth century, but the complete texts depend on the White Book and the Red Book, and Guest worked chiefly from the Red Book, which is held at Jesus College, Oxford.