Brand's autoattack particle is the same as Morgana's but orange instead of purple.
Brand's appearance was likely inspired by Ignus from Planescape: Torment.
Brand comes from Proto-Germanic *brandaz,meaning "fire, torch, sword (poetic)." In English, brand only means "torch" and "sword" in literary contexts, unlike its related Germanic cognates which still mean "fire".
His host's given name Kegan is likely based on Old Norse kagi "young tree, shrub", cf. English keg and key; the surname Rodhe likely from hróðr "glory".
Pyroclasm derives from pyroclastic, a term used to describe fast-moving volcanic material, from Greek πῦρ "fire" & κλαστός "broken"
Brand might have been based on the mythical Norse fire giant Surtr "swart(y)": Brand resembles him (blackened skin with lava crackling underneath) and, like Surtr in Ragnarök, Brand is destined to renewing the world by burning it.
Brand occupying the body of seafaring marauder Kegan Rhode might be referencing Human branding, a process in which, for example a slave, was branded a searing mark that would then leave a permanent scar.
Brand is the 'being of living fire' Nasus and Renekton fought right before the fall of Shurima. The details of his imprisonment are unknown but he was released and used by Xerath as a distraction for stealing the Ascended power meant for Azir.
Lokfar being a region of the Freljord implies Brand's icy imprisonment many centuries ago was at the hands of the Frostguard, going so far as Lissandra having a taunt specifically directed at Brand.
↑G. Kroonen, Etymological Dictionary of Proto-Germanic, p. 277, 284