The 7.7 magnitude quake struck about 70km (43 miles) off Kirakira, according to the US Geological Survey.
It said it occurred at about 17:40 GMT (04:40 Friday local time), with a 5.5 magnitude aftershock about 25 minutes later.
Places were being evacuated, with people moving to higher ground.
Users on social media said there were some blackouts in the capital, Honiara, but that there were no immediate reports of damage.
Some said there had been no tsunami sirens.
Hazardous waves could hit parts of the coasts of the Solomons, Vanuatu, Papua
New Guinea, Nauru, New Caledonia, Tuvalu and Kosrae within the next three hours, the Pacific Tsunami Warning Center said.
US officials say they are investigating if there is a threat to Hawaii.