Griffiths, 20, came through the youth ranks and has had spells with Runcorn Linnets and Marine.
He joins former Prescot team mates Jack Lenehan and Chris Almond at Broadhurst Park.
FC United have also signed left-back James Joyce from Marine.
Joyce, 24, played in the Europa League for Llandudno in 2016/17, and made 25 appearances for the Mariners last season.
Teenage defender Markell Foulds, 18, has signed from Salford City.
Meanwhile, Jan Palinkas has joined Wythenshawe Town for the 2019/20 to help with his development.
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