In the Qu'ran, Mohammed is clearly against the act of suicide, no matter the reason.
Recently, I came across a statement that the people who changed this line of thinking where Ayatollah Khomeini and Hafez al-Assad. Which...Doesn't make any sense because majority of terror groups are Sunni and I doubt they care what Shia leaders say.
What made this method acceptable?
From my understanding the use of suicide attacks are not truly suicide in the sense that we see it (self murder) and more of the suicide attacks we hear about in war action movies (the goal is so important that we are going to send you on a mission you have little chance of surviving). There's a lot of this that is less historical and more explanation based on Homeland Security training and analysis of events. Note my use of "martyr" is purely academic. I don't believe for one second that any terrorist perverting their religion will receive any reward in the afterlife.
When Pan Am Flight 103 was blown up over Lockerbie Scotland, the man who placed the bomb onto the aircraft did not board the aircraft. In response, and I'm not 100% when, it became a rule that in order for your bag to fly on a flight your ticketed for, you must also fly (the obvious exception is if the airline misplaces your bag). It also create the baggage screening system that was in place and utilized on September 11, 2001. The same trend of attackers surviving their attacks continued through most of the 90's with terrorists like Ted Kaczynski (Unibomber), Timothy McVey (Oklahoma City Bomber), Ramzi Yousef (1993 WTC and The Bojinka Plot) all carrying out attacks and walking away from their explosives. Yes, I know only one was Islamist, but the point is that the biggest attacks in the late 90's weren't suicide attacks.
With each attack, authorities instituted defenses against such attacks making them harder to carry out. Soon it would have become apparent that leaving a package unattended would lead to a plot being foiled, delivering the explosive directly to the target would have to be the way to ensure success.
Now, that's not to say that suicide attacks are products of a necessity found only in the late 90's onward. the 1983 bombings of the Marine Barracks and the US Embassy in Beirut were carried out by suicide bombers and Andrew Kehoe may have used the first suicide truck bomb in 1927. But looking at the trend in attacks and defense, the need for martyrdom has risen. It is much more difficult for someone to leave a bomb in a crowded market than it is for them to walk in and blow themselves up. Following the 9/11 attacks Richard Reid wore shoes with TATP and PETN packed into the cavities in the rubber soles and was ready to blow himself up.
Now, obviously the masterminds of such attacks are not usually the ones to become a martyr for the cause. They often employ women, children, the mentally ill, the physically infirm and the "young and stupid" devotees to the cause to carry out the attacks for them (http://abcnews.go.com/WNT/IraqCoverage/story?id=608103). These people occupy classifications of people who are not routinely expected of harming people or are so new to the mission as to not be on any watchlist. In 2004, Chechen rebels sent two women on board two separate aircraft with powerful explosives built into their bras and downed two Russian airliners killing 90 people. The reason for this was they knew the Russian authorities would balk at patting down the bras of innocent looking women. Umar Farouk Abdulmutallab, the "Underwear bomber" copied the tactic by cramming a device into his "tighty whities", knowing that Dutch authorites wouldn't grab his junk and find the bomb. Of course the September 11th attacks display the ultimate in suicide attack using trained and willing "sacrificial warriors" (I'm trying to define them as they saw, not how I see because its not very nice how I describe them), turning four aircraft into flying incendiary bombs. I heard once, and I can't remember where so take this for what its worth, that not all of the 19 hijackers knew they were committing a suicide attack. The use of these unaware attackers is why TSA pats down grannies, kids and people wheelchairs. Its also why a same-gender-as-you TSA screener will be getting almost too friendly with you and why they invented the backscatter machines. If we're afraid of checking the underwire of a woman's bra, then that's where the next bomb will be hidden.
Prior to my leaving federal service, there was a return to remote attacks not requiring a martyr. In 2010, Saudi intelligence allowed the UK and UAE to intercept devices that were simply explosive material packed into printer cartridges with detonation systems attacked. In the movie The Hurt Locker, IED's were left on street corners and in public squares. The San Bernadino shooters did not appear to necessarily want to martyr themselves after the first attack, but obviously saw it as an option if stopped by police. They didn't kill themselves, but fell during a shootout with police.
So all this info leads me to believe that no one person, or even two, led the cause of suicide bombers. This was a natural progression from having to carry out an attack with nearly insurmountable defenses. They are defined by us, the victims as suicide attacks because we see suicide as morally wrong. They see themselves as divine sacrifices for the greater good or simply as heroes willing to pay the ultimate price for the groups victory. But it is still the case that whenever possible, attacks are carried out where no martyr is required so as to retain the individual for future operations.
(I hold a Certificate of Completion of the Homeland Security program at American River College (a DHS/TSA supported program. College newspaper story with my bald head on the right and the Catalog page on the program) as well as 5 years service as a TSA Transportation Security Officer. This link takes you to a slideshow I created for a presentation in one of the HLS classes if you want a little more info on how terrorism affected US transportation regulations. There may be historical events or people who have advocated for suicide attacks that impacted its use, but this is how I learned and how I interpret the evolution of Islamist terrorist tactics over time.)