I'm trying to find an example of Frodo's cloak's effectiveness. I dimly remember reading an occasion when Frodo is trying to sleep, cold and vulnerable. It's a passive use, where the cloak works 'as needed', this time to keep Frodo warm.
You might be remembering this scene in Mordor as Sam and Frodo start on their crossing after coming down from the Tower of Cirith Ungol
He shivered a little. 'What I really need is something warm,' he said. 'It's gone cold, or else I've caught a chill.'
'You can have my cloak, Mr. Frodo,' said Sam. He unslung his pack and took out the elvencloak. 'How's this, Mr. Frodo?' he said. 'You wrap that orc-rag close round you, and put the belt outside it. Then this can go over all. It don't look quite orc-fashion, but it'll keep you warmer; and I daresay it'll keep you from harm better than any other gear. It was made by the Lady.'
Frodo took the cloak and fastened the brooch. 'That's better!' he said. 'I feel much lighter. I can go on now
The Return of the King, Book VI: Chapter 2, The Land of the Shadow
The only other mention of the cloaks' warmth is in the initial description from the "leader of the Elves"
But they should serve you well: they are light to wear, and warm enough or cool enough at need. And you will find them a great aid in keeping out of the sight of unfriendly eyes, whether you walk among the stones or the trees.
The Fellowship of the Ring, Book II: Chapter 8, Farewell to Lórien
I can't find a specific mention of using the cloak to stay warm when sleeping, but while they are walking through Mordor, Frodo has to remove much of his armour and heavy gear due to his fatigue. He mentions being cold, and Sam gives Frodo his cloak, implying that it will keep him warm.
‘What I really need is something warm,’ he said. ‘It’s gone cold, or else I’ve caught a chill.’ [...]
‘You can have my cloak, Mr. Frodo,’ said Sam. He unslung his pack and took out the elven-cloak. ‘How’s this, Mr. Frodo?’ he said. ‘You wrap that orc-rag close round you, and put the belt outside it. Then this can go over all. It don’t look quite orc-fashion, but it’ll keep you warmer; and I daresay it’ll keep you from harm better than any other gear. It was made by the Lady.’