The old lady oddly admits: "I give up" (5)
I believe the answer is:
yield
'give up' is the definition.
(I've seen this before)
'the old lady oddly admits i' is the wordplay.
'the old' becomes 'ye' (resembles a historical spelling of 'the').
'oddly' means one should take alternating letters (letters in the odd-numbered positions are used).
'admits' indicates putting letters inside (to admit is to allow in).
The alternate letters of 'lady' are 'ld'.
'ye'+'ld'='yeld'
'yeld' placed around 'i' is 'YIELD'.
(Other definitions for yield that I've seen before include "Produce as harvest or surrender" , "Supply" , "Cave in" , "Abandon" , "Make a submission" .)