Old English leaþr "foam, soap, washing soda," from Proto-Germanic *lauthran (source also of Old Norse lauðr "washing soap, foam"), from PIE *loutro- (source also of Gaulish lautron, Old Irish loathar "bathing tub," Greek louein "to bathe," Latin lavere "to wash"), which is from root *leue- "to wash" + instrumentative suffix *-tro-.
The modern noun might be a 16c. redevelopment from the verb. Meaning "violent perspiration" (especially of horses) is from 1650s; hence the transferred sense "state of agitation" (such as induces sweating), attested from 1839.
from a Middle English variant of letheren (v.), from Old English leþran (late West Saxon lyþran) "become covered with (sweat, blood, etc.)," also transitive, from Proto-Germanic *lauthrjan (source also of Old Norse leyðra "to clean, wash;" see lather (n.)). Meaning "to form in froth, produce suds or foam" is from c. 1600. Related: Lathered; lathering.