First off.
Global Average temperature isn't a meaningless term. If used correctly, it's a useful tool, irrespective of how it's calculated, and yes, there are several ways of calculating it.
Firstly, presumably it's based on the average daily temperature - in otherwords, measure the temperature over a 24 hr period, and compute the average, this is a good thing because it will take into account any changes in the average night time temperature. There's a couple of main ways of doing this, you could measure the temperature at hourly intervals and then perform a simple mean calculation, or, you could measure the temperature continuously, find the area under the graph, and then compute the temperature which divides that area in half. Both are equally valid. To some extent, doing the first one is a lower resolution version of doing the second one (essentially it represents a
Rieman Integral IIRC). The first measurment is easier to perform, the second measurement is more accurate.
Secondly, it is possible to calculate a global average temperature, and again, there are a number of ways of doing this, but all of them involve using the daily average temperature at a number of locations. Like with all statistics, the more information you have, the more accurate the inferences you can make. With modern satellite technology, itis possible, or nearly possible to get a continuous read out of the surface or air tempterature at every point on the earths surface, from this you can perform much the same technique as the second method I mentioned. You can, for example, map out the isotherms, and then compute the thermal equivalent of a mean radius. Or, if you have a bunch of stations placed around the world, as we have had for a number of years, you can collect the data, and combine them, but, this requires the assumption that the physical location of the thermometer is in an area that is representative of the region that it's measuring for - For example, you're not dealing with a thermometer that's in a greenhouse in Iceland, and calling that the average temperature of Iceland.
This is the part where things get tricky, because there are a bunch of things that can influence a thermometer that don't neccessarily represent local trends, or global trends.
If these thermometers happen to be on an evenly spaced grid, then you can simply calculate the average of the average daily temperatures and be done with it. However, it's far more likely that thermometer distribution is going to reflect population distribution, it's going to be clumpy and uneven (for example, I'd wager there are more NOAA weather stations on the eastern sea board of the US then there are in the american mid west). So what you need to do then is calculate a weighted average, assigning each average daily temperature a value depending on how much surface area (or some such thing) the thermometer is representing.
Now for my pet peeve. Statistics is not dishonest. It's only when people misapply it (and that happens frequently) that it starts to become questionable - and that's only as a direct result of people not understanding the implications or the short comings of the data they're dealing with.
Most statisticians would probably tell you that it isn't neccessarily wise to look at average temperatures alone, ideally you should probably also be looking at the global peak average temperature (the global average of the daily maximum temperatures, or nightly minimum temperatures) and maybe a percentile or two. The reaosn behind this is that a shift in temperature might simply be a shift in modality, or a change in the shape of the distribution, without a corresponding increase in the spread of the data.
And that's one of the key points that needs to be looked at - does the shift in average data correspond to a shift in peak data, or a change in spread.
The information that I have seen on trends in global climate change is that not only is the average going to increase, but the extremes are going to become more extreme - the record setting cold snaps are going to be colder, and possibly more frequent, and the record setting heat waves are going to be hotter and possibly more frequent - so we're seeing an increas in the variance of the data, as well as an increase in the mean (or we will if these models are right).
Unfortunately for everyone, Isotope proxies can only give information about averages, because that's what they respond to.