Earlier I used a 0.5W straw hat LED, but it is not so efficient and the lumens/watt is around 70-80. 

So I am planning to replace the LED with a 0.5 Watt cool white SMD LED ( 2835  package). As per the SMD LED datasheet, the forward voltage (Vf) is 3.2V and the luminance is 150 lumens per watt. 

My target lumens for the different modes of operation are as follows:

1. Mode-1 ( Low Brightness ): 15 Lumens with a backup time of 24hrs

2. Mode-2 ( Medium Brightness ): 30 Lumens with a backup time of 10 hrs

3. Mode - 3 ( High Brightness): 70 Lumens with a backup time of 5hrs

The power required to drive for the above modes is as follows:

1.  Mode-1:  15/150 = 0.1W

2. Mode-2:  30/150 = 0.2W

3. Mode-3:  70/150 = 0.467W

In the following table, calculations are presented to estimate the battery capacity.  I have considered a 18650 Li-Ion battery (3.7V) with 80% DOD. 

Formula Used in the above table:

1. LED Current (mA) = Power (W )/ LED Voltage (3.2V)  

2. Energy (Wh) = Power (W) x Backup Time (Hrs)

3. Updated Wh = Wh/ Efficiency 

4. Capacity (mAh ) = Updated Wh / Battery Voltage ( 3.7V )

5. Final Capacity =  Capacity / Battery DOD

From the above table, the maximum charge required for the battery is 928.1 mAh. However considering the market availability and taking some additional margin for sunny days, the final battery size selected is 1200 mAh.