The pretzel's name is about arms, not prayers. Whatever a monk may or may not have done with leftover dough, the word itself points to the Latin for "arm."
Etymonline traces English pretzel to German Prezel or Brezel. That came from Middle High German brezel and Old High German brezitella. Behind those sits a medieval Latin baking word, given as brachitella by Etymonline and bracellus by the OED. Both are built on bracchium, "arm," which Latin took from Greek brakhion.
The OED, as quoted on the linguistics blog Language Log, says the biscuit was "so called on account of the resemblance to folded arms." Etymonline is more careful and calls the folded-arms shape a presumption. Either way, the name describes the knot.
That leaves the famous story. In its usual form, an Italian monk in 610 AD crossed strips of dough to look like hands folded in prayer. He gave the baked knots to his students to encourage their learning and called them pretiola, "little rewards." Wikipedia reports this only as the most popular story. Another source it mentions moves the monastery to southern France.
The pretiola part is also not where the word comes from. Dictionaries derive pretzel from the arm word, not from a Latin word for a reward. The legend may still echo something real, since a knot of folded limbs is what the etymology describes.
The firmer history is visual. A 12th-century manuscript from Alsace, the Hortus deliciarum, may contain the earliest picture of a pretzel. In southern Germany the pretzel has served as the emblem of bakers and their guilds since at least the 12th century.
English picked up the word late, in 1836. Why the English spelling starts with a "p" is its own small puzzle. The OED suggests English speakers heard the soft, unaspirated "b" of regional German speech as a "p." Soft pretzels are recorded by 1893 in the German-speaking parts of Pennsylvania. By 1945 "pretzel" could mean anything badly twisted.