What are the advantages of diluting a stock solution?

The obvious advantage of diluting a stock solution by adding water, is that you will make the stock go further. 

If you have made your own stock, which you can do really simply by boiling up a pan of vegetables and/or the leftover bones from a joint of meat, a good idea is to freeze this liquid, and then when you want to use it, you can dilute it with boiling water (which will also defrost the the ready made stock).

Flavour-wise, an advantage to diluting stock is that it makes the flavour less intense. This is good if you are using subtle flavours and do not want your dish to be overpowered by the flavour of the stock. 

A good point to remember is that the longer you cook a dish for, and the more the liquid reduces (thickens), the stronger the flavours will be.