Name Meaning and Origin:

The Fausett name (including  its many variants) originates from Northern England and is a “topographical” or “toponymic surname” name deriving from the Anglo-Saxon Old English (pre 7th Century) word’s “fah” and “side”, meaning “brightly coloured”, “variegated”, or “flowery” on the “side” or “slope” of a hill.

fáh

fáh, fág; def. se fága, seó, ðæt fáge; adj.
Coloured, stained, dyed, tinged, shining, variegated; tinctus, cŏlōrātus, vărius, versicŏlor, discŏlor
Bleóbry; gdum fág shining with variegated colours, Exon. 60 a; Th. 218, 9; Ph. 292.

síde

an ; f.
II. side of a house, ship, etc.
Duru ðú setst be ðære sídan (the side of the ark ), Gen. 6, 16: Past. 22; Swt. 169, 24.
III. marking direction on this or that side

II a. the side of a hill :– Of þǽre wídgyllan sídan þæs muntes e devexo montis latere, Gr. D. 112, 19.