As you probably appreciate, the definition of "suicide bombing" isn't always historically obvious. (E.g., would Japanese kamikaze count?) But let's only look at the systematic, strategic use of high-explosive suicide bombs for the purpose of "terrorism," e.g. not as a form of "normal" state warfare (with a full acknowledgment that "normal" and "state" can be problematic terms in this context), and not some random, chance occurrence.

OK. Within those bounds, my understanding is that the first major application of suicide bombing is in the 1980s, by Hezbollah. They venerate Sheik Ahmed Qassir as the "first suicide car bomber" who killed and inured over 140 Israelis in an attack on their headquarters in Tyre in 1982, "the deadliest single disaster in Israel's history." Strictly speaking, a Hezbollah driver who attacked an Iraqi embassy in late 1981 may have been the first to utilize the strategy, but they pay more credit to Qassir because the entire thing was recorded.

Hezbollah followed this strategy up over the course of the 1980s with a lot of success, particularly the attack on the US Marine Barracks in Beirut in 1983, where a dump truck was full of 12,000 lbs of high explosives and tanks of propane was used to murderous effect.

The strategy migrated from Hezbollah to the Sri Lankan "Black Tigers," a wing of the Tamil Tigers, in the late 1980s, who were impressed by the "lethal effectiveness" of the Hezbollah methods (and their ability to generate huge amounts of attention and fear). The Tamil Tigers are credited for inventing the "suicide vest" approach, where instead of using a vehicle you just load up an individual with explosives. Hezbollah and the Tamil Tigers trained together in the 1980s.

Recommended reading on this grim topic is Mike Davis, Buda's Wagon: A Brief History of the Car Bomb. Super interesting stuff, and Davis is a provocative guy (his earlier book, City of Quartz: Excavating the Future in Los Angeles, is considered a must-read). Davis sees suicide bombing as an offshoot of car bombing in general, one that only applies to groups with very specific types of religious beliefs (he notes that Afghan resistance fighters — and I don't think he is including all mujahideen here — did not do this and would not have because their religious beliefs prohibited suicide). Davis thinks the re-appropriation of anonymous, civilian vehicles (wagons, trucks, cars, planes, etc.) into military weapons is the interesting historical trend to follow, and he starts off with the 1920 Wall Street Bombing as the origin point of this trend.