I need compact design which is light in weight and small in size.
I have searched on internet for rechargeable 5V battery but didn't find a solution.
There are mostly 3.6V,7.4V,11.1V rechargeable battery solutions available.
Actually I have a box (size is nearly equal to a cabinet drawer) which will be locked automatically through solenoid.
Box is small and there is no much space to keep large design there.
For that I need a small battery with charging module.
If it won't drain the battery too much, an integrated boost converter and LDO for the 5V output and a dedicated charger IC for recharging - maybe not small nor a module... Sparkfun Adafruit, etc., may or may not have a module that does what you need that is small.
My cabinet drawer is made up of acrylic and small in size. USB power banks are heavy for this.
For that I need Battery based power supply which should be compact
I am using 5V 1A Push pull solenoid for lock
The USB pack that Klaus has suggested, will be the simplest, highest capacity, lightest weight and lowest cost you can find.
They come in all sizes, and are sold in the web by almost everyone.
They also comply with the USB charging and output ports.
You shouldn't dismiss Klaus suggestion without providing any meaningful data point.