Lasagna is probably named after the pot, not the pasta. Its most widely cited ancestor is a Greek word for the stand a cooking pot sat on.
Etymonline gives the path. Greek lasanon meant a "pot with feet" or a trivet. Latin borrowed it as lasanum, "a cooking pot." From that came Vulgar Latin lasania and then Italian lasagna, plural lasagne. The name slid from the vessel to what was cooked in it.
There is a less appetizing branch. World Wide Words notes that the older Latin lasanum is "said to" have also meant a chamber pot. The site calls these the word's "insalubrious origins." Wikipedia also lists "chamber pot" among the senses of the Greek word.
Italian etymologists, World Wide Words says, prefer a different Greek source, lagana. Wikipedia describes that rival theory, which traces the word to laganon, a flat sheet of dough cut into strips. Greek still uses lagana for a flat, unleavened bread. That theory makes lasagna a name for the pasta itself rather than its pot.
Italian records of the dish are old. Wikipedia cites a ballad copied by a Bolognese notary in 1282 as the oldest written reference. The chronicler Salimbene di Adam mentioned lagana cum caseo, lasagna with cheese, in 1284. World Wide Words gives the first Italian mention as 1281. The first recipe, in the early 14th-century Liber de Coquina, describes boiled sheets of dough sprinkled with cheese and spices.
English is where the sources split. Etymonline says the dish was known by the late 14th century as loseyne or losenges, a "dish of cheese and large noodles." Wikipedia notes that the loseyn in the 14th-century cookbook The Forme of Cury resembles lasagna in layering and name. World Wide Words disagrees. It argues that loseyns is an old form of lozenge, and that "lasagne" is first recorded in English in 1760.
Either way, the modern dish is newer than the word. Tomatoes only reached Europe after 1492. Etymonline says early recipes used the same ingredients as period recipes for macaroni, and it calls the tomato version a 19th-century creation. The modern spelling, it adds, appears in Italian dictionaries by the 1590s.