grade school science experiment peltier module and led
Peltier modules are VERY inefficient compared to the absolute max thermodynamic efficiency. You MAY get about 20% of theoretical max. 

ABSOLUTE maximum possible efficiency relates to the delta temperature drop and the absolute temperature of the hot side in degrees Kelvin.
T_Kelvin ~= TCelsius + 273.   

MAX theoretical efficiency is (Thot-Tcold)/Thot.
Thot is in degrees K.
(Thot - Tcold) = delta_T  can use K or C degrees as the difference is what matters.      

In degree K (Kelvin)
0 C = 273K and
100C = 373 K. 

With Thot = 100C = 373 K (boiling water) and
Tcold about 30 C then
Delta T = 100-30 = 70. absolute available efficiency is 70/373 = maybe 18%
and actual = about 4%. 

BUT when using a hand
delta T is VERY small and
energy flow is very very small and
energy out is very very very small indeed.   

Using eg very hot water and cold water as Thot &amp; Tcold you MAY light an LED.
Using your hand you have almost no chance. 
Alas.

Try this.
Place a voltmeter across Peltier leads.
Place Peltier module on a block of ice.
Place your hand on "hot" side.
Observe voltage change (if any) as your hand warms the hot side.
This is voltage with no current.

Now, IF you saw ANY voltage:
Change to the lowest available current range.
Some multimeters have 1.999 mA ranges so they can theoretically "resolve" 1 uA changes.
Repaeat above experiment and note current.

You now have the open-circuit voltage and short circuit current.
Operating voltage and current will be rather lower in each case.

Power out will be significantly less than Voc x Isc.

IF you have Voc x Isc >= 500 microwatts then you MAY be able to get a glimmer from a very efficient LED. 

A very efficient modern white LED is nicely visible at about 3 mW. (3V x 1 mA).
At this level light out is not too useful - maybe 0.5 lumen. Enough to light an area about 50mm to 100mm square to very minimal reading level. 

To check that you are not "fooling yourself" with your measurements, after taking a reading, turn the Peltier over and swap hot and cold sides. The polarity of the readings should reverse and the numerical results should be about the same once the system settles down again thermally.



I did some quick 'playing' here with a Peltier module intended to be operated as a heater/cooler on 12V. Using boiling water and ice I only managed about 400 mV - too low for an LED. Short circuit current was encouraging - much higher than I expected at tens of mA. A number of these in series should operate an LED - or one using an energy harvester - but that is probably more complex and expensive than you wish. Try experimenting with 1 x Peltier as above first and report the results you get and we can advise on possible alternatives. Even a hand plus ice produced measurable output - trying it with an energy harvesting IC as Spehro suggested would be interesting. He suggested the LTC3108 Ultralow Voltage Step-Up Converter and Power Manager - which would have been my suggestion too, but he said it first.  They are not a cheap IC - but similar to Peltiers in cost. ( $6.28/1 at Digikey, in stock ).
If you were enthused and wanted to reduce cost you could try emulating their 'front end' which is a simple oscillator with a high ratio step up transformer and feedback. 

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The only slightly cheaper but easier to implement LTC3105 - 400mA Step-Up DC/DC
Converter with Maximum Power
Point Control and 250mV Start-Up may be of interest.  The use of a simple inductor rather than a transformer makes implementation easier and the 225 mV minimum startup voltage is easily within the range of the Peltier's output. 

  



DIY Energy Harvester?

This is the LTC3108 block diagram,with the portion inside the red line being what would need to be built to achieve the actual energy recovery. Adding the blue portion is a bonus. 
  In a DIY version starting the oscillator without battery or other voltage to provide startup biasing requires a MOSFET with an EXTREMELY low Vgsth (= turnon voltage). Few parts qualiy and even those are marginal. One which may is the BSH103
N-channel enhancement mode
MOS transistor - $0.36/1 from Digikey - and they have 40,147 in stock :-).As shown the oscillator diagram is schematic only and would need investigation. The typically 100:1 transformer should be available from other applications.