A strange physics question.

I did this a while ago and then questoned it.

I wanted an ice cold protein shake and the blender was broken so I used some ice cubes. I wanted to make the drink cold as quickly as possible.

My first thought was, I'll put the ice cubes in the shake and leave it in the freezer for 10-15 mins.

Then I thought maybe this would be slower, as although the freezer is cooling down the water, the ice wouldn't melt as fast.

I now think leaving it out would be faster. I know the difference would be minute and not something worth thinking about, but now I'd like to know which would be quicker?

Isn't there some weird thing that if you put hot water in the freezer it freezes quicker than cold? 

For the record, I'm purely interested on which would be faster, not for other ideas on making it cold lol.

If a body at temperature A is put in contact with a body at temperature B such that the temperature of the body A is higher than that of temperature B, then heat energy will flow from body A to body B until the temperatures of body A and body B are equal and no heat energy flows between them.
 
If you just put the ice cubes in the drink, let's say they're exactly 0 and your drink is 20 degrees. Then upon equilibration your drink is 18 degrees (say) and the ice cubes have melted.
 
If you put the whole lot in the freezer, at -15 degrees, we'll assume your freezer acts as a temperature resevoir, that is, its temperature doesn't change no matter what you put in the freezer, then your drink will keep cooling until its temperature is the same as the freezer, which in this case is -15 degrees.
 
So the freezer will undoubtedly make your drink cooler. You basically answered your own question when you said the ice cubes wouldn't melt as fast in the drink in the freezer. Why? Because the drink will be cooler...
 
In terms of speed, if you consider some simple models such as Newton's law of cooling or Fourier's law, they're proportional to the temperature difference, and so the larger the temperature difference, the faster the cooling. Therefore the freezer wins again.
 
Physics gets a bit more complicated when things change phase (ice to water to steam and vice versa) but in general the above holds.

The ice cubes will melt faster at room temp, but the overall temp of your ice + drink mixture will be higher than if you stuck it in the freezer.
 
Imagine the situation where your ice cubes are in your drink but don't melt at all - how can that happen? Well, it can happen if your drink is the same temperature as the ice initially i.e. 0 degrees.
 
So the fact your ice cubes melt less is exactly what you want to achieve.