Although I don't really know where the etymology of "Years of Lead" comes from, nor am I entirely certain that the phrase had an application outside of Italy, I can offer some insight into the very complex relationship that the PCI with the Soviet Union, and how it changed in the 1970s.

The Italian Communist Party (PCI) was the spiritual successor to the Communist militias which had taken up arms against the Fascist Regime between 1944 and 1945. Although these militias were composed of existing anti-fascist activists as well as returning soldiers/deserters, they had received small albeit perceptible amounts of aid from the Soviet Union. Thus "Sovietization" of these militias, and by extension the "sovietization" of opposition to the dying regime, definitely was tangible; but I would not call it complete. A particular anecdote I'm very fond of is the way Hemingway recalls the mood in Italy (especially in the north) in Across the River and Into the Trees: the owner of Harry's Bar in Venice has bought an equal number of cases of Whiskey and Vodka, not knowing who would liberate Venice first!

In the early days of the Republic of Italy, it's clear that the majority members of the Italian Communist Party leadership had a favorable view of the USSR: the General Secretary Togliatti had a direct line to Moscow. However, as early as the 1950s it became difficult for the PCI to justify increasingly evident oppression in the Eastern Block, culminating in the Hungarian Revolution of 1958. Togliatti began advertising "The Italian Way Towards Socialism" in the 1958 election, signaling that the pro-USSR faction within the party had definitely become the minority. By the 1968 Prague Spring, the PCI was actively condemning the Soviet Union. By 1976, right in the middle of the "Years of Lead," the General Secretary of the PCI Enrico Berlinguer declared in a high-profile interview in the Corriere della Sera newspaper that his new policy of “Eurocommunism” was unconditioned by the regimes behind the Iron Curtain, and that at any rate he felt safer living under the umbrella of NATO than he would be in a Warsaw Pact country. A shocking pro-American declaration, although Berlinguer wasn't by any means supportive of most US policies in Europe and the world.

Enrico Berlinguer's Eurocommunism was a way for the Italian Communist Party (Partito Comunista Italiano, or PCI) to once and for all shed the guise of a vanguard revolutionary party and become a more moderate and appetizing potential governing coalition partner for the Christian Democracy Party, whose pluralities were becoming increasingly slim. Although these overtures to forming a joint government were certainly born out of an urgency to resolve the country's economic and political instability (in no small part catalysts for the Years of Lead) I have always been swayed by the argument that Berlinguer was also incentivized by the death of the Chilean President Salvator Allende over the course of his overthrow by a Military Junta in September 1973. Allende was a Marxist, and had been implementing policies of nationalization and collectivization similar to those proposed by the Italian Communist Party. The alleged role that the US had played in supporting the overthrow of Allende left Berlinguer unwilling to find out if the US would back the army in a bid against him, especially as the Christian Democrats were losing votes left and right and a social climate where his party was being associated with political violence.

When the 1976 election came along, the PCI’s new platform was rewarded with 34.37% of votes, their best showing yet. The DC by some miracle continued to hover around 38%. DC kingpin Giulio Andreotti — prodded by the left wing of the party headed by Aldo Moro — was obliged to assemble a “Government of National Solidarity,” and the PCI agreed to abstain from votes of confidence. The event was dubbed the “Historic Compromise” and this arrangement would remain in place with Andreotti at the helm until 1979. However, cause and consequence are very confusing at this point in time: Berlinguer and the PCI scored a political victory that had repositioned the party, but this galvanized extremism: 1976 saw the start of a new wave of terrorist attacks. As you probably very well know, kidnappings and shootings targeted politicians, judges and prosecutors as well as businessmen and police. Most dramatically, in March of 1978 the Red Brigades kidnapped and subsequently murdered former Prime Minister Aldo Moro, the head of the left wing of the Christian Democracy Party notable for his sponsorship of the “Historic Compromise.” His body was found strategically laid halfway between the Communist Party headquarters and the Christian Democratic Party headquarters.

Moro’s kidnapping and murder was a catastrophic failure of Italian institutions on all levels and shocked the country. But in the 1979 elections, voters would choose to punish above all others the PCI. The “Historic Compromise” had promised economic stimulus and social stability, but they had seen the opposite: fiscal austerity and an increase in extremist terrorism. A new wave of "Radical" parties also emerged positioning themselves even farther to the left than the Communists! However, the PCI did not revert to its old line; perhaps thanks to the strength of Berlinguer's convictions and ability, and with the more moderate Socialist Party even obtained an unsuccessful mandate from President Pertini to explore possible governing coalitions.

Confidence in the PCI would rebound as the Christian Democrats' new coalition under former Minister of the Interior Francesco Cossiga, maintained a controversial and ineffective hard line against radical terrorism: 1980 saw 125 lives claimed by radical terrorism, the blackest year yet in Italy’s fight against extremism. Not two years later, a financial scandal would prove the definitive catalyst to elevate the first non-Cristian Democrat to the office of Prime Minister.

Prior to his removal from the office of PM, Cossiga's coalition had looked for validation in international affairs: Italy wholeheartedly supported NATO’s deployment of Pershing II and Gryphon mid-range missiles in Europe. However, it would seem that the PCI had successfully decoupled itself from conversations about the Soviet Union, and structured their opposition on the narrative around the government's inability to manage inflation and economic indecisiveness.

Lastly, it's worth concluding with the observation that in spite of Berlinguer's official line, there remained a backbone of pro-USSR administrators in the Italian Communist Party such that when Berlinguer suffered a fatal stroke at a rally in 1984, the Communist Party leadership sought refuge in pro-Soviet nostalgia, only to lose any ability to form coherent policy after Michail Gorbachev’s reforms within the Soviet Union gained steam.

Edit: if you'd like to read more, I can suggest

Gilbert, Mark. The Italian Revolution: The End of Politics, Italian Style? Boulder: Westview. (the first few chapters)

Ginsborg, Paul. A History of Contemporary Italy: Society and Politics, 1943-1988. London, England: Penguin, 1990. (although Ginsborg is a bit too dismissive of the DC as a legitimate political force, characterizing them as little more than corporatist goons)

And if you can read Italian: Berlinguer, Enrico. La Crisi Italiana: Scritti su Rinascita. 1985. Rome: L’Unità.