Hello.

what does "bocugum" means?

I´m an insect ?

That´s only my try

Maybe it is someone referring to someone else as "my insect (pig? hog?)"

It is a silly word that is used to  show her affection to someone. It is commonly used among lovers . Especially some women call their lover/hubby as " böcüğüm" "[ My [little] bug "

Personally  I would never want to be called that

You can hear that silly word alot in the tv serious called " Yahşi Cazibe".

Hmm... if the word for "insect" is actually böcü, why don´t you add only -m as the possessive suffix saying böcüm (like to other nouns ending with a vowel: annem, kedim...). Böcüğüm really sounds like "I am an insect" to me 

the word is "böcük" which is the kid´s or some lovers way to say for "insect". The proper word "böcek".

So  it is ==>        böcük --> "k" changes into "ğ" ---> böcüğ + üm ---> böcüğüm --> My bug 

Progressive assimilation.

Another possibility is that could be a missing letter in "boncuğum"

boncuk --> bead

boncuğum ---> my bead, my bead-liked eye. , My blue eyed , My beloved blue eyed lover.

this nick name is sometimes given to "cats and dogs"., however some people use that word to call other people who has blue eyes just to show their afffection to them.

aha thanks!