Oh, I see now.
I confess that I also thought he died in 1808.
There are, it seems two Adye graves in Gibraltar.
John Willet ADYE, son of Captain Ralph Willet Adye, died 20th March 1804, aged 2 years 1 months, laid to rest in the Garrison Chapel
Ralph Willet ADYE, Captain and Brigade-Major in the Royal Regiment of Artillery, died 22nd October 1804, aged 34 years, a victim to the distemper raging in the garrison
I think Willet/Willett (I've seen both variations) is a given name but am not sure. There are a number of quite well known Adye's in the RA, some quite senior, as late as the Boer War at least. None share the name Willet. I assume they are the same family though I have nothing other than name and regimental association to support the assumption.
As for the spelling of the surname, Adye seems right, though, for what it is worth, Capt James Cook, the RN officer and explorer, has his name written as 'Cooke' in contemporary records sometimes, including his father. The explanation for such variations is often that the name was written phonetically.