Data:

(1) Liz/Lizzie/*Elizabeth
(2) Liz/Elizabeth Coppock
(3) Elizabeth/*Liz/*Lizzie Edwards Coppock
(4) *Edwards (Coppock)

LFG Solution:

NAME	→	FIRST	(MIDDLE)	(LAST)
		↑=↓	(↑OFFICIAL)=c+	(↑OFFICIAL)=+
FIRST	→	Liz
FIRST	→	Lizzie
		(↑OFFICIAL)=-
FIRST	→	Elizabeth
		(↑OFFICIAL)=c+
MIDDLE	→	Edwards
LAST	→	Coppock

HPSG Solution (Thanks to Ivan Sag!):

     _______name_______
    /	      	        \
m-name           ___independent-name__
(Edwards)       /             /       \
        first-name       l-name      complex name:
	/     \          (Coppock)    |FIRST f-f-name        |
 i-f-name     f-f-name                |REST <(m-name,)l-name>|
(Liz, Lizzie) (Liz, Elizabeth)