I checked the Library index for St James Church and found this extract:
"The church was not assigned a Parish until 1844". This probably accounts for his burial here, even though he seems to have lived in Town (and married at St Nicholas 1768).
http://archive.liverpool.gov.uk/dser...83%20JAM%27%29
The first naming of Penny Lane, found so far, is the 1840s well after his death ((1799), or that of his eldest son (also James Penny d.1820).
A trawl of early newspapers would seem to be the most likely place to find street-name changes and announcements about creating new streets etc. I'd hate to be the one searching for the word 'penny', though.