c. 1300, romen, "walk, go, walk about;" early 14c., "wander about, prowl," a word of obscure origin, possibly from Old English *ramian "act of wandering about," which is probably related to aræman "arise, lift up."
There are no certain cognate forms in other Germanic languages, but Barnhart and Middle English Compendium point to Old Norse reimuðr "act of wandering about," reimast "to haunt."
"Except in late puns, there is no evidence of connexion with the Romance words denoting pilgrims or pilgrimages to Rome ...." [OED], such as Spanish romero "a pilot-fish; a pilgrim;" Old French romier "traveling as a pilgrim; a pilgrim," from Medieval Latin romerius "a pilgrim" (originally to Rome). Transitive sense is from c. 1600. Related: Roamed; roamer; roaming.