From Encyclopedia of Heraldry by John Burke and John Bernard Burke (1844)
Okay, this should start looking familiar! By the time of Henry II, the three lions we know and love today were well established. There are a few potential sources for the lions; the most popular origin story I’ve seen has Henry adopting the two lions of Normany via his mother, Empress Matilda, and adding another lion (the arms of Aquitaine) when he married Eleanor. There is not a whole lot of proof for this theory, but it is a pretty common approach to arms, especially in the early days of heraldry, before systems of quartering, dimidiation, and cadency came into use. (Unsurprisingly, the Burkes go with this theory; it’s got more of a narrative shape to it.)
The other possible source is that Henry’s three lions are a modified version of the arms borne by his father, Geoffrey Plantagenet. Geoffrey’s funeral effigy is one of the oldest examples of European heraldry in existence, dating to around 1160. It depicts him bearing azure (probably) six lioncels or. It’s certainly possible that Henry swapped out the tinctures and continued using his father’s well-known gold lion. This could be a bit circular; a late 12th-century writer whom the Burkes identify as “John, the Monk of Harmonstier, in Tourain” recounts that Henry I gifted Geoffrey with “an image of golden lioncels hung around his neck” (very rough translation). The Burkes do reproduce this quote in the original Latin, but don’t necessarily draw the connection between Geoffrey and Henry II. Not that I’m necessarily surprised; it’s not uncommon for English heraldic writers of this period to just kind of ignore the whole “Anarchy” business.