How to select FR-4 board thickness
Unless I have specific space requirements I always go for the thickest they offer (1.6mm).  It sounds like iTead?  At least it's the same range they provide for the same cost ;)

There are a number of factors that can influence your board thickness though, but some can be ignored for "normal" (i.e., non-RF) designs:


Space - Thinner boards take less space in your packaging. You'll often find very small devices (those tiny USB dongles, bluetooth headsets, etc) will use a much thinner board to save space.
Connections - Using PCB edge connectors, or PCB trace based USB A connectors, etc, require the PCB to be the right thickness to fit in the mating portion of the connector.  
Impedance Matching - Every two (or more) layer board will act like a capacitor between traces on adjacent layers.  The thickness of the board defines the thickness of the dielectric, and hence the value of the capacitance.  For high frequency signals where impedance matching is critical that capacitance has to be taken into account, and getting the right thickness of board for your design can be quite troublesome sometimes.
Flexibility - Yes, thinner boards can flex. That can pose problems for P&amp;P machines if the board is too big, or you have it panellised wrong.  Vee-grooves are not a good idea with very thin boards, and small panels are better than large, so it's down to the fab house what they set as an upper limit on size for those.  Also boards that flex can cause broken connections with components (especially BGA), so a thin board will need good support from the casing it's in to avoid that.
Component compatibility - Some throgh-hole components are designed for specific board thicknesses. For example, press-fit connectors.
Weight - Thinner boards are lighter and can be cheaper to ship and also result in lighter end products. 


I think that's all the main concerns, but if anyone can think of any others please feel free to add them.