It seems to indicate that the person who died was believed to have assets of some kind, but as far as the curate and churchwardens knew, there was no will/probate and no administration. However, I can only speculate as to why it was thought necessary to make a note of this in the register: possibly as a way of indicating that the next of kin had been advised to obtain the relevant grant, but that their failure to do so was not the curate's fault?
The entry wouldn't be anything to do with the burial, but as probate was handled by church courts, the curate may have thought it was his business to know and record such things; and as the Bishop's Transcripts were submitted to the authorities each year, this may have been his way of mentioning the situation without making a formal report about it.
You may or may not be aware of the North East Inheritance Database - an index of pre-1858 probate records for the Diocese of Durham, with links to the FamilySearch images in many cases. You can search this by name, place, date etc (in Advanced Search), so you might find it interesting to compare the entries for Tweedmouth at that time with what's in the BT. You can find it at:
http://familyrecords.dur.ac.uk/nei/data/intro.php
