"Finnechty cup" is an expression that comes from the area on the Moray Coast in the North East of Scotland surrounding the village of Findochty, "Finnechty" being the local pronounciation of Findochty.
Apparently, it was the perception of the inhabitants of the neighbouring villages that the folk in Findochty were rather ungenerous when pouring out a cup of tea! Hence the saying came into being and any cup of tea anywhere which was not deemed to be full enough was referred to as a fine "Finnechty cup". I don't know when it originated, I first heard it when visiting my in-laws in Buckie in the 1970's but was never able to find out why Findochty folk had such an awful reputation for being mean with the tea!! I am pretty sure that it is still used in the area, it is certainly used in our family whenever the level of tea in a cup is not quite full enough, so hopefully will still be in use in 5 years time.....