Putting together some of the sources provided in the comments:
There were indeed four Soviet diplomats abducted in Beirut in 1985. One was shot and the other three were freed about a month later.
About two months after the three other diplomats were freed, the story re. the family member of one of the kidnappers surfaced. So the story is quite old, which of course does not yet make it true.
This report makes it sound as if the liberation was instead brought about by a lot of diplomacy (which, just as the other story, may or may not be true):
Embassy First Secretary Vladimir Beskournikov could not conceal his joy and laughed heartily when he was reached by telephone to confirm the release.
"They are free," he said. "They managed to make it with our help and the help of all our friends. They are safe and in good health but they are tired and must have a rest."
"There are so many friends who assisted us, so many organizations," he said. Asked if Syria had been involved in the hunt, he said, "Yes, yes of course," but he refused to give details of how the Soviets' captors had been persuaded to let them go.
This article has some more info on who (according to the soviets) was involved in the freeing of the Soviet hostages.
Re. the kidnappers: Hezbollah is a Shia organization that is very open about being allied with Iran (e.g. by the choice of its logo). It was not the only Shia organization active in the Lebanese civil war, another important one was Amal, which was backed by Syria and therefore friendly with the Soviets.
The group that kidnapped the hostages called itself the "Islamic Liberation Organization". Several newspapers report that that was a Sunni group, while Hezbollah is Shiite. However, at least one newspaper says that the Islamic Liberation Organization was connected to another organization named "Islamic Council of Ulema", which may be identical to the "Union of Muslim Ulama" described as allied with Iran here.
In 1985, Iran was still involved in the Iran-Iraq war (in which the Soviets sold lots equipment to Iraq), and there was also the Soviet war in Afghanistan, which is directly east of Iran. Therefore it is not unlikely that Iranian-backed Shiite militias in Libanon would have quite different interests from Syrian-backed Shiite ones.
So IMHO that the kidnappers were part of Hezbollah (as a shorthand for "Some Lebanese faction allied with Iran") is quite plausible, but not entirely clear.
I think it is worth pointing out that the "family member abducted" story has a number of variants. This website claims that the Russians threatened to also bomb Qom or Teheran
We aren't only talking about people in Beirut. I'm talking about Tehran and Qom [Shiite holy city and the residence of Ayatollah Khomeini], which is not that far from Russia's borders. Yes, Qom is very close to us and a mistake in the launch of a missile could always happen. A technical error, some kind of breakdown. They write about it all the time. And God or Allah forbid if this happens with a live, armed missile.
It also claims that the Russian diplomat who was killed was "riddled with machine gun bullets", which is contradicted here, and that the Hezbollah family member was shot.
(as a side note, Qom is actually not closer to the former Soviet Union or Afghanistan than Teheran is. The Soviet Republics that were closest to Iran were Armenia, Azerbaijan and Turkmenistan, not the RSFSR).
A source that can be found on google books (The Secret War with Iran by Ronen Bergman, without page numbers) claims that the family member was not shot, but killed with a knife. This page claims that the Hezbollah family member was actually killed by someone else and the Soviets just happened to stumble across his body. Both sources also claim that Qom is really close to the former Soviet border, which I find a bit strange.