Leaflet 1.0.0 has a new built-in L.tooltip class that depreciates the Leaflet.label plugin. The tooltip points to the center of the shape and does not move with the mouse.
L.polygon(coords).bindTooltip("my tooltip").addTo(map);
Demo: https://jsfiddle.net/3v7hd2vx/91/
To address the OP comment about the tooltip displayed in the center of the polygon, which may not be visible when the polygon is very large and the current scale is high, you can use sticky :
L.polygon(coords).bindTooltip("my tooltip", { sticky: true
Updated demo: https://jsfiddle.net/3v7hd2vx/402/
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