You could connect the map's "onMouseMove" event to a setTimeout function that would show the popup. Also, use the clearTimeout to stop the countdown whenever the mouse is moved. Here's some example code.
function onMapLoaded(map) {
var timeOutAction;
dojo.connect(map, "onMouseMove", function (e) {
// reset the previous timeout that would call for the popup.
if (timeOutAction !== undefined) {
// if timeOutAction is not null or undefined, clear it out.
window.clearTimeout(timeOutAction);
}
// use setTimeout to run the function that shows the popup after 1 second.
timeOutAction = window.setTimeout(function () {
/* perform identification of map using point e.mapPoint */
}, 1000); // 1000 milliseconds = 1 second.
});
// when the mouse moves off the screen, reset the timeout function.
dojo.connect(map, "onMouseOut", function (e) {
window.clearTimeout(timeOutAction);
});
}