I don’t know, that’s what the guy told me: “Our name used to be Ibn-xxxx, but it meant...”
Ibn- is ‘son of’. Ibn- become bin- when occurring further down the line in a name. I cannot recall the proper grammar rules though. But that's where the ‘bin-’ in ‘Bin-Laden’ comes from, ‘Son of Laden’.
Ibni would be ‘My son’.
Anytime you see ‘Abd’ beginning a name it connotes slave or servant. The name ‘Abdul’ means ‘servant of’ or ‘slave of’. Abdullah translates to “Servant of Allah”.
‘Son of a slave’ would probably be ‘Ibn abdul’ or something, I think. But don’t take that to the bank.