An attempt at home — eager at heart to become permanent (7)
I believe the answer is:
abiding
'permanent' is the definition.
(similar in meaning)
'an attempt at home eager at heart' is the wordplay.
'an' becomes 'a'.
'attempt' becomes 'bid' (bid is a kind of attempt).
'at home' becomes 'in' ('I'm in' can mean 'I'm at home').
'at heart' says to take the centre.
The centre of 'eager' is 'g'.
'a'+'bid'+'in'+'g'='ABIDING'
'to become' is the link.
(Other definitions for abiding that I've seen before include "Dwelling; enduring, lasting" , "standing" , "Putting up with: long-lasting" , "Remaining or residing" , "Stopping" .)