I want to make a portable device that's using a battery, i'm thinking about using 9V battery.
With that single battery I will be using that for 1 hall effect sensor, 1 arduino nano, 1 LCD 1602, 1 op-amp LM324N
Is using a 300mah rechargeable 9v battery good choice? or is there any better choice?

 

Is using a 300mah rechargeable 9v battery good choice?

Depends upon the total current draw of your circuit and how long you want the circuit to operate before the battery is discharged.
Need both of those parameters before I can answer your question.

 

Is using a 300mah rechargeable 9v battery good choice? or is there any better choice?

They're only useful for low current applications. How much current do you plan to draw?

 

That namo alone draws 19-24ma from what I am reading. If the LCD is backlit, you might want to design your circuit so the backlight times out after a period.