Solar system battery sizing

For BATTERY SIZING, I would continue as below.
(1) For example, Led lamp of 12 V 12 W, ON for 6 hours per day. average sunshine hours is 5 hours.
(2) Ah discharged in lamp = (12 W/12 V)x 6 = 6 Ah.
(3) LED driver efficiency, = 90% ( assumed figure ), therefore LED driver input Ah drawn from battery = 6/0.9 = 6.67 Ah.
(4) Battery discharge efficiency = 90% (approx figure). Therefore Battery INPUT Ah = 6.67/0.9 = 7.4 Ah.
THUS PV MUST be able to deliver ATLEAST 7.4 Ah PER DAY to GLOW the LED LAMP for 5 hours. In addition to this loss in interconnecting cable & ADDITIONAL capacity to RECHARGE completely discharged battery within 7 days ( approx. ) should be considered.
(5) Battery capacity will depend on HOW MUCH OF BACKUP TIME is required, to be delivered by battery WITHOUT CHARGING, before it is COMPLETELY DISCHARGED. Lets say design requires 3 cycles of 6 hours each.
Thus Ah capacity required = 7.4x3 = 22.3 Ah.
(6) Battery is generally NOT discharged to 100% (called Depth of Discharge or DOD in short). Generally upto 70% DOD is acceptable, for approx. 3-4 years of practical battery life..
Thus Battery Ah capacity should be 22.3/0.7 = 31.75 Ah.
(7) This capacity is NAMEPLATE capacity of battery at 27 deg. C. Thus temp. compensation has to be provided if battery is to be operated at lower temp. If min. ambient temp. is 5 deg. C, then approx. 15% additional capacity is required because AVAILABLE battery capacity falls with reduction in temp.
Thus battery capacity should be 31.75 x 1.15 = 36.5 Ah.
(8) FINALLY round it off to the nearest higher size battery available. Say 42 Ah.
All this is at 12 V.

For PV sizing, I would proceed as below.
(1) 42 Ah Battery has to be charged within 7 days of 5 sunshine hours = 5x7 = 35 hours.
(2) Battery charging efficiency = 90% ( approx ) Ah required to be delivered into battery = 42/0.9 = 46.67 Ah
(3) Ah required per day = 46.67/7 = 6.67 Ah.
(4) Ah required by LED Lamp = 7.4 Ah
(5) Cable loss @2%
(5) Total Ah required per day = (6.67 + 7.4)x1.02 = 14.35
(6) Current required for 5 sunshine hours = 14.35/5 = 2.87 A

Thus select a PV module which can deliver 2.87 A for 5 hours.
Typically 12 V 35 W module will do the job.
Select 12 V 42 Ah Battery. ( I have considered Sealed maintenance free battery)

Finally, and very important consideration, COST!!! If this cost is not a SELLABLE preposition, then reduce battery capacity & PV module rating. Accordingly battery backup time will reduce & battery recharging time will increase.