Griffiths, 20, has spent six years at Everton, but his contract is due to expire at the end of the season.
Described by Everton as having a "strong and commanding presence", Griffiths made 29 appearances on loan for Cheltenham this season.
This follows temporary stints with Northwich and Halifax in recent years.
Griffiths arrives to challenge Craig Samson for the number-one jersey as Brill exits after failing to make a first-team appearance.
The former Inverness Caledonian Thistle goalkeeper is understood to be on his way to Colchester, where he trained for a few days earlier this month.
Motherwell have signed Everton goalkeeper Russell Griffiths on loan as Dean Brill departs.