From Forebears.io:
'the holmer,' one who lived on a holm or home (v. Holm); compare Bridger, Fielder, or Churcher.
or from Occup. 'the heaumer,' a maker of helms or helmets
From 23 AndMe:
1. English: topographic name for a dweller by a holly bush, from Middle English holm (Old English holen) + the agent suffix -er, a type of formation characteristic of Hampshire and Sussex. See Holm 2.
2. English: habitational name from Holmer (Herefordshire), Holmer Green (Buckinghamshire), or Homer (Devon), or else from other unrecorded placenames formed with Middle English hol(g)h + mere ‘pool by or in the hollow’ (Old English holh + mere).