Selection of only monthly data on Google Earth Engine?

You nee to use ee.Filter.calendarRange(start,end,field) to filter by all Januaries from 2000 to 2010.

field could be:

• field (String, default: "day_of_year"): The calendar field to filter over. Options are: 'year', 'month', 'hour', 'minute', 'day_of_year', 'day_of_month', and 'day_of_week'.

So, the code is:

var imageCollection = ee.ImageCollection("LANDSAT/LT05/C01/T1_TOA");

// Use ee.Filter.calendarRange to filter by year and month

var img = imageCollection.filter(ee.Filter.calendarRange(2000,2010,'year'))
.filter(ee.Filter.calendarRange(1,1,'month'));

// reduce image collection with mean()
var mean = img.mean();

// Set visualization parameters
var vizParams = {
  bands: ['B3', 'B2', 'B1'],
  min: 0,
  max: 0.5,
  gamma: [0.95, 1.1, 1]
};

// add layer
Map.addLayer(mean,vizParams);

This code process specifically your task description, doesn't consider cloud mask or a cloud coverage filter.