what does "an old remnant of powers granted to the American executive"mean?
Arliss snorted. Kelsea glanced at him and found him grinning around a newly lit cigarette. It smelled just as terrible as the other one had, but Kelsea said nothing. It was Arliss who’d informed her of the obscure right of the crown to take direct military action, an old remnant of powers granted to the American executive. When she met his eye, he gave her a wink.

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It's in reference to a set of antiquated rights that were bestowed upon the leaders of the country. Coming from a time when things were very different than the world of today.

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The 'crown' refers to the 'the Crown.' It should be spelled with a capital latter. The Crown was the British monarchy, and the powers residing within that monarchy. At the time it was King George III. The monarchy of Britain presumably had a right to order direct military action (? without the consent of parliament).

When the colonials (the Americans) won their independence from Britain they inherited some old British laws it seems. The laws are probably as ancient and obsolete as they are in Britain itself.

But we need a historian to tell us more.

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It means it is an old law that is rarely or never used. When the USA was formed the world was different and the government might have powers that they wouldn't have if it was set up now.