Lack of care showing, ultimately, in name chosen (7)
I believe the answer is:
neglect
'lack of care' is the definition.
(I've seen this before)
'showing ultimately in name chosen' is the wordplay.
'ultimately' says to take the final letters.
'in' indicates putting letters inside.
'name' becomes 'n' (common abbreviation e.g. on forms).
'chosen' becomes 'elect' ('elect' as an adjective can mean chosen).
The last letter of 'showing' is 'g'.
'n'+'elect'='nelect'
'g' inserted into 'nelect' is 'NEGLECT'.
(Other definitions for neglect that I've seen before include "Omit by carelessness" , "Pay inadequate attention to" , "Fail to act or take care" , "Lack of proper care" , "Disuse" .)