From the Online Etymology Dictionary:
O.E. matte, from L.L. matta "mat made of rushes" (4c.), probably from Punic or Phoenician (cf. Heb. mittah "bed, couch")
While we might not associate a mita מיטה with a mat, the sense of "mat" actually preserves the original meaning more than "bed".
The word mita, according to Klein is:
From נטה (= to stretch out; to incline, bend). For sense development, compare Greek kline (=bed) which is related to klinein (=to cause to slope, slant, incline).
(Two English words that are connected to each of the Greek are clinic and incline).
Stahl points out that the Biblical mita was a blanket or rug that was spread out on the floor - a "mat". The difference between the mita of old and the mita of today might be seen in the Talmudic mourning practice of כפיית המיטה - "turning the bed over". The later halachic sources say that we do not keep this custom of mourning because our mita is different - it's much easier to turn over a mat than a bed.
Mita in Talmudic times developed the meaning of "family, offspring" - as in a description of Yaakov - מטתו שלמה - mitato shleima, "his (conjugal) bed was perfect", i.e. "his children were all righteous".
From the root נטה we get two other words that are spelled the same as mita (when it is spelled without a yod, as it does in the Tanach). The word מטה can also mean mata - "downward, down" and mateh - "stick, rod, staff".
Based on this confusion (the original text had no vowels), there are different readings of Bereshit 47:31. The Masoretic text has עַל-רֹאשׁ הַמִּטָּה "al rosh hamita" - "at the head of the bed". However, the Septuagint translates it as "over the head of his staff" - in other words "al rosh hamateh".