I decided to look into this matter and whilst I can't find any definite proof, there does seem to be a fair amount of circumstantial evidence.
First, I'm not the only one to have spotted the link between the name of the Twin Peaks character and title of the later film, M Morse on chud.com in his comprehensive review of Twin Peaks has this to say of the character:
I’m also admittedly intrigued by Nicky’s background, which involves his
parents being killed, and eventually the possibility that ‘Little Nicky’ (Anyone else instinctively think of that stupid Adam Sandler movie whenever Nicky’s referred to that way?) is malevolent
More significant perhaps is the links between Adam Sandler and former actors and actresses of Twin Peaks, including David Duchovny and Francis Bey.
In May 1995, when Duchnovny hosted Saturday Night Live, Sandler played numerous roles, so it seems reasonable to assume they had met. Duchovney had of course appeared in several Twin Peaks episodes as DEA Special Agent Dennis/Denise Bryson.
Further in Adam Sandlers 1996 hit, Happy Gilmour, Canadian actress Francis Bey played his grandmother. She had previously appeared in a Twin Peaks episode as the mysterious Mrs Tremond. She also starred in the film 'Fire Walks With Me' and has worked with Lynch on other films such as 'Blue Velvet'.
Other actors link Sandler to Twin Peaks in the other direction, namely they appeared on film or tv with Sandler before appearing in the series or the film 'Fire Walks With Me'. These include Keifer Sutherland and Billy Zane.