Claw Isle is an island in the Narrow Sea, just off the shores of Crackclaw Point on the east coast of Westeros. It is located north-east of Dragonstone. It is part of the Crownlands and it is currently unclear who it holds fealty to, following the death of Stannis Baratheon.
In the books
In the A Song of Ice and Fire novels, Claw Isle is the seat of House Celtigar. It lies a few hours sail from Dragonstone.
Similar to House Velaryon of Driftmark, the Celtigars were followers of House Targaryen when it originally fled from Essos to avoid the coming Doom of Valyria. They are therefore of Valyrian descent. These refugee Valyrian families survived on islands in Blackwater Bay for another century: the Targaryens on Dragonstone, the Velaryons on Driftmark, the Celtigars on Claw Isle, and possibly others. During the War of Conquest they provided troops to their Targaryen liege-lords, which made up the small original core of Aegon the Conqueror's army.
According to Salladhor Saan, the castle in Claw Isle is reputedly stuffed with Myrish carpets, Volantene glass, gold and silver plate, jeweled cups, magnificent hawks, a sea eagle that Lord Celtigar had trained to fly from the wrist, an axe of Valyrian steel, a horn that could summon monsters from the deep, chests of rubies, and more wines than a man could drink in a hundred years. Though Lord Celtigar had shown the world a miserly face, he had never stinted on his own comforts.
After Lord Celtigar bent his knee to Joffrey Baratheon, Axell Florent and Salladhor Saan planned to raid Claw Isle to exact retribution for Lord Celtigar's defection, taking advantage of the fact that at present the isle is weakly held by women and children and old men. Axell intended to destroy the isle population completely, to make the realm see the fate of those who bed with Lannisters. Stannis tended to approve the plan, commenting that the risk was small, and though the isle itself is worthless, its fall would serve notice to Tywin that Stannis's cause is not yet done.
Axell warned Davos that if he objected to the plan, unpleasant "accident" might happen to him. But Davos, conscientious as ever, firmly objected, calling the plan "folly and cowardice". He claimed that Stannis had no enemies on Claw Isle: Lord Celtigar indeed bent the knee to Joffrey, but he came when Stannis summoned him and his men fought for Stannis and were killed for him. Claw Isle was now weakly held by women and children and old men because their husbands and sons and fathers died on the Blackwater fighting beneath Stannis's banners. It would be cruel and unjust to swoop down on the homes they left behind, to rape their widows and put their children to the sword, as Axell suggested.
Stannis accepted Davos's opinion and cancelled the plan. When he and Davos were alone, he admitted it was wrong, and only Lord Celtigar would be punished for his betrayal. Axell resented Davos, but did not carry out his threat against him. Salladhor was disappointed the raid was cancelled, for he coveted the treasures Lord Celtigar kept in his castle (especially the magical horn), but he and Davos remained friends.