Everyone agrees a mortgage is a "dead pledge." The harder question is what died.
The word is an Anglo-Norman compound of mort, "dead," and gage, "pledge." David Wilton at Wordorigins.org says it was modeled on medieval Latin mortuum vadium. He cites a 1293 Anglo-Norman legal record in which one man hands land to another "en morgage" for a sum of money.
English borrowed it as morgage. Its first known English use, Wilton says, is figurative. In John Gower's Confessio Amantis, written before 1393, a marriage vow "lith in morgage." The legal sense appears in English around 1400. The spelling picked up its silent "t" in the mid-1500s, added to look more like the Latin.
The most quoted explanation of the "dead" comes from the jurist Sir Edward Coke (1552-1634). Etymonline quotes his commentary on Littleton. If the borrower fails to pay, the land is taken "for ever, and so dead to him." If he pays, "then the pledge is dead as to the tenant," meaning the lender. Either way, something dies.
That account is neat, but it may be a later rationalization. The American Heritage Dictionary points to an older answer. Around 1189, a legal treatise attributed to Ranulf de Glanvill described two kinds of pledge. In a mort gage, the creditor kept the land's income without reducing the debt. In a vif gage, a "living pledge," that income paid the debt down.
On this reading the pledge was "dead" because it did no work toward repayment. Because Christian rules barred profiting from lending, the dictionary says, this arrangement was treated as a kind of usury. By Thomas Littleton's treatise of 1481, the mortgage had changed shape: the borrower kept the land, under a condition. Coke's explanation fits that later form.
French itself gave the word up. Etymonline notes that modern French uses hypothèque instead.
The second half of the word has relatives in everyday English. Old French gage, "pledge, pay, reward," had a northern French twin, wage, which gave English wage around 1300. The first half shares its Latin root with mortal, from mors, "death."