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Why? Both seem to be used, with Lake Geneva being used more commonly. Are there good reasons for preferring one over the other?

I think there are good reasons to prefer "Lake Léman".

The lake is split between France and French-speaking part of Switzerland. In French, the name is Léman. I do think using the local name is better, even more when this local name is acknowledged (en.wikipedia shows both Geneva and Léman).

To me, born and raised in Romandy, "Lake Geneva" feels like "Tokyo Country" for Japan or "California Ocean" for the Pacific.

As is true of many place names in Switzerland, it doesn't have one single name.

Lake Léman = Lac Léman = Genfersee = Lake Geneva

There is nothing wrong with the title of the Nature paper: Giant Lake Geneva tsunami in AD 563

Curiously, I can't get Google maps to show a name for the lake, unlike several other Swiss and Italian lakes in that zone.

May be the reason.