We present two new primary mechanisms for the synthesis of the rare nucleus 9Be, both triggered by nu-induced production of 3H followed by 4He(3H,gamma)7Li in the He shells of core-collapse supernovae. For progenitors of ~8M_sun, 7Li(3H,n_0)9Be occur
s during the rapid expansion of the shocked He shell. Alternatively, for ultra-metal-poor progenitors of ~11-15M_sun, 7Li(n,gamma)8Li(n,gamma)9Li(e^-anti-nu_e)9Be occurs with neutrons produced by 4He(anti-nu_e,e^+n)3H, assuming a hard effective anti-nu_e spectrum from oscillations (which also leads to heavy element production through rapid neutron capture) and a weak explosion (so the 9Be survives shock passage). We discuss the associated production of 7Li and 11B, noting patterns in LiBeB production that might distinguish the new mechanisms from others.