Mouse (n.) Old English mus "small rodent," also "muscle of the arm," from Proto-Germanic *mus (source also of Old Norse, Old Frisian, Middle Dutch, Danish, Swedish mus, Dutch muis, German Maus "mouse"), from PIE *mus- (source also of Sanskrit mus "mouse, rat," Old Persian mush "mouse," Old Church Slavonic mysu, Latin mus, Lithuanian muse "mouse," Greek mys "mouse, muscle").
Plural form mice (Old English mys) shows effects of i-mutation. Contrasted with man (n.) from 1620s. Meaning "black eye" (or other discolored lump) is from 1842. Computer sense is from 1965, though applied to other things resembling a mouse in shape since 1750, mainly nautical.
Mat (n.1) loosely joined natural materials used as bedding, etc., Old English matte, from Late Latin matta "mat made of rushes" (4c.), probably from Punic or Phoenician matta (compare Hebrew mittah "bed, couch"). Meaning "tangled mass" is from 1835. That of "piece of padded flooring used in gymnastics or wrestling" is attested from 1892; hence figurative phrase go to the mat "do battle" (1910). The Latin word also is the source of German Matte, matze; Dutch mat, Italian matta. French natte "mat, matting" is from Late Latin secondary form natta (compare napkin). (http://etymonline.com)