Add an event listener on a Marker in Leaflet

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There should be no special difficulty in attaching a callback to marker(s) click event. You would simply use myMarker.on('click', callback) like you did with the map. You would also have to do that for every marker you want to attach a callback to.

Another possibility would be to add all your markers into a Feature Group (e.g. simply instantiate your markersLayer with L.featureGroup() instead of L.layerGroup()), so that you can attach the callback directly to that Group. It will receive the click events from individual markers, and you can access the individual clicked feature using event.layer.

var markersLayer = L.featureGroup().addTo(map);

// populate map from stops…

markersLayer.on("click", function (event) {
    var clickedMarker = event.layer;
    // do some stuff…
});

Demo: http://jsfiddle.net/ve2huzxw/74/

Similar question asked by someone else on Leaflet forum: https://groups.google.com/forum/#!topic/leaflet-js/RDTCHuLvMdw

Modify your map population loop to assign properties to your marker.

//populate map from stops
for (var i in stops) {
  var oneMarker = L.marker(L.latLng(stops[i].Position.Lat, stops[i].Position.Lon), {
    title: stops[i].Description
  }).bindPopup("<b>" + stops[i].Description + "</b>").openPopup();

  oneMarker.properies.name = stops[i].Name;
  oneMarker.properies.description = stops[i].Description;
  oneMarker.properies.othervars = stops[i].othervars;
  oneMarker.addTo(markersLayer);
}

Later, to access these properties (feature properties as they're called) in the onclick event,

markersLayer.on("click", markerOnClick);

function markerOnClick(e) {
  var attributes = e.layer.properties;
  console.log(attributes.name, attributes.desctiption, attributes.othervars);
  // do some stuff…
}

The properties.var approach has the added benefit of making your marker be in the standard GeoJson format. Makes it compatible if, say, you need to export the data as shapefile, import markers from shapefile, etc.