Why not the Palestinians, or the Lebanese, or the Dutch, or the Papua New Guineans? What place did Libya occupy in the popular American consciousness in the mid-1980s?
During the 1980s, Libya (having tried and failed to buy nuclear weapons from other countries) tried to get access to uranium and plutonium alike for the purpose of building weapons. It completed a research reactor (with Soviet help) back in 1979, that gave it the opportunity to produce plutonium, and evaded IAEA safeguards during that time. They also attempted to buy nuclear material from black market sources throughout this period.
You can read more about their plans for nuclear weapons at the links below:
http://www.nti.org/e_research/profiles/Libya/Nuclear/index.html
Back to the Future was released in the United States on July 3, 1985 and according to various draft scripts released online, the screenplay was largely written in 1984.
If you haven't seen the movie, it contains the following lines:
"Doc, you don't just walk into a store and-and buy plutonium! Did you rip that off?"
"Of course. From a group of Libyan nationalists. They wanted me to build them a bomb, so I took their plutonium and, in turn, gave them a shoddy bomb casing full of used pinball machine parts. Come on! Let's get you a radiation suit. We must prepare to reload."
In the first half of the 1980s, one of President Ronald Reagan's significant foreign policy dilemmas was the issue of Libya. David Willis, in his 2004 book, called Reagan's dealings with Libya "the first war on terrorism," and while I don't agree that it was the "first," it certainly was a war on terrorism.
In 1969, Muammar Gaddafi led a coup against King Idris of Libya. After the 1973 Cultural Revolution, Gaddafi consolidated his power through the introduction of sharia law and the manipulation of Islamist sentiment that surged following the Yom Kippur War. Gaddafi brutally purged opponents of his regime and conducted distractionary wars against his neighbors.
His government's ideology shifted and flowed throughout his tenure as dictator, sometimes in a confusing muddle, but it maintained its Islamist flavor as a unifying force. Gaddafi had a great interest in assuming a prominent role on the international scene. He was an active participant in pan-Arab movements such as the Federation of Arab Republics and created the Islamic Legion as a tool to unify the region.
This desire to be a dominant force led him to support terrorist and "liberation" movements throughout the Middle East and Africa -- even the IRA received some support. Libyan support is believed to have been behind the 1972 Munich Olympics terrorist attack, the Rome and Vienna airport attacks, the bombing of Pan Am Flight 103, and dozens of other high-profile terrorist attacks.
There was even a direct confrontation with the United States in 1981. The Gulf of Sidra incident happened because Libya claimed a large chunk of the Mediterranean as its territorial waters, in violation of international agreement. The U.S. Sent a carrier air group into this space, and it was challenged by Libyan aircraft. On Aug. 19, 1981, Libyan jets fired on American aircraft and were shot down in turn.
President Reagan took a hard line against the Libyans for their support of terrorist causes, because of their support of international conflict, their backing from the Soviet Union, and the Gulf of Sidra incident. Libyan diplomats were expelled, sanctions levied, and other diplomatic and military moves were taken.
Since the 1969 coup, Gaddafi had taken special care to pursue nuclear weapons. In the 1970s, for example, he attempted to purchase an atomic bomb from China. He worked with the Soviet Union to obtain atomic assistance, moves that accomplished the construction of an atomic reactor in 1981.
Libya pursued Pakistan and India diplomatically, again seeking atomic assistance. They acquired centrifuges and other materials, and were known to be pursuing nuclear weapons in the early 1980s, at the time the screenplay was written.
To summarize, Libya was thought of in the United States at the time of the writing of the screenplay as a violent, rogue state seeking nuclear weapons, making it a perfect foil for the start of the movie.
Ironically, the use of Libya in the film became much more appropriate in 1986, when it was still playing in some theaters around the world. In that year, the confrontation between America and Libya again became violent, with both naval and aerial clashes following terrorist attacks against American targets in Europe.