If mayhap looks to you like a relative of its synonym perhaps, you're right—the words are related. Both ultimately derive from the Middle English noun hap, meaning "chance" or "fortune." Mayhap was formed by combining the phrase "(it) may hap" into a single word (the word maybe, another synonym of mayhap and perhaps, was developed similarly from may and the verb be). Hap in the phrase is a verb essentially meaning "to happen," and the verb hap comes from the noun hap. Perhaps came about when per (meaning "through the agency of") was combined directly with the noun hap to form one word. Today, mayhap is a rare word in contrast with the very common maybe and perhaps, but it does show up occasionally.
Examples of mayhap in a Sentence
mayhap I could see you again next week? I so enjoyed our date tonight