The missile hit the home of Kamil Yasin in the southern town of Hula early on Tuesday, killing the owner’s grandchild, said security sources.
The dead boy’s eight-year-old brother was also injured.
The origin of the projectile is still unclear. But witnesses said an Israeli warplane was seen hovering over the border area after it struck.
The boy’s death came hours after an Israeli sergeant died on the Lebanese border.
Two other soldiers were reportedly wounded.
Israel said Hizb Allah resistance fighters were responsible for the death of its soldier. But the Lebanese resistance group had denied its forces were involved.
The managing editor of Lebanon’s al-Safir newspaper Mutaz Midani pointed out that Hizb Allah would never deny a resistance attack if it had been responsible for it.
The death of the Israeli sergeant may have been caused by an individual angry with the Jewish state’s attacks in the occupied Palestinian territories and Syria, Midani told Aljazeera.
Israel raised tensions again on Tuesday, firing shells into the occupied Shiba farms region without causing injuries, said Lebanese security forces.
The shells fell about 100 metres away from the UN-demarcated Blue Line along the Lebanese-Israeli border.
Members of the United Nations Interim Forces in Lebanon (UNIFIL) carried out patrols along the line as Israeli reconnaissance drones hovered overhead.
On Monday, Israeli troops opened fire and hit a bus travelling along a road near the border between the villages of Adasiah and Kfar Kila. There were no casualties, said Lebanese police.
Hizb Allah was formed as a resistance group in the early 1980s following the Israeli invasion of Lebanon in 1982.
Fears are mounting that Israel
It ousted Israeli forces from south Lebanon in May 2000 following a 22-year-occupation.
The missile-related death and cross-border shooting followed an incident on Monday, when a lorry belonging to UNIFIL was hit by gunfire, agencies reported.
“A UNIFIL water truck was hit by three bullets from the Israeli side,” a UN official told Reuters. “All our trucks are clearly marked.”
Lebanese security sources also said Israeli troops had opened fire across the border earlier in the day. But Israel denied its troops were involved.
The border incidents come amid rising regional tension, just a day after Israel launched its deepest air strike into Syria for 30 years, attacking what it claimed was a training camp for Islamic Jihad fighters.
The resistance group claimed responsibility for a bombing of a restaurant in the Israeli town of Haifa on Saturday that left 19 people dead.
Islamic Jihad denied having military training bases in Syria.
The Israeli airstrike has attracted widespread international criticism and sparked worries about regional stability.