I don't anything know about a beeper on a FX888D
Nothing mentioned in the manual or pics of the PCB. Hakko is good quality, assuming you have a genuine one.
I would check the 5VDC power, that's about all you need to boot up. Even with the wand unplugged the display should come up. Also would do a visual on the PCB for bad solder joints or flux corrosion.
You could try reset to factory default settings, but it also erases the calibration
In order to reset the HAKKO FX-888D back to the factory default settings,
"... Hold UP and ENTER buttons and turn power on.
The display shows units 'A' (for Asian defaults and display in Celsius) or 'U' (for US defaults and display in Fahrenheit). UP button to toggle the value shown between 'A' and 'U'.
When you have made your selection, press the ENTER button and the factory reset will be complete..."
From:
http://kb.hakkousa.com/KnowledgebaseArticle10599.aspx