Part of big issue to be raised in old court — increased rent (8)
I believe the answer is:
octuplet
'part of big issue to be raised' is the definition.
(issue can mean someone's offspring)
'old court increased rent' is the wordplay.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'court' becomes 'ct' (abbreviation used in road names).
'increased' becomes 'up'.
'rent' becomes 'let'.
'o'+'ct'+'up'+'let'='OCTUPLET'
'in' is the link.
(Other definitions for octuplet that I've seen before include "One of eight in birth" , "One of eight born together" , "One of a group of eight children born at the same time" .)