>>13837664연(緣) consists of 糸(string) and 彖(decide). According to etymology of 緣, 彖 was only added to change the pronunciation. Basically, 緣 character represents the invisible "string" that connects people. How did we go from string to "string that connects people"?
인연(因緣) is a Buddhist concept that represents relationship that forms between people or having a connection to an object.
In Buddhism, 인 is a direct force that brings forth a result and 연 is an indirect force that helps 인. An example I found is in using rice plant as a metaphor. Rice grain itself is 인 while sunlight, water, dirt, and temperature are 연.
You can see how 인 and 연 is a single package and is the process that reaches causality.
Also in Buddhism, causality depends on karma.
So when she says 연이 닿는다면 다음에 또 봐요 (닿는다면 means if they reach and 다음에 또 봐요 means see you again), then what she really means is 인연이 닿는다면 or (skipping few steps here) "if our karmic relation reaches each other".
You can also think of it as "if that invisible string connects us then we'll see each other again" or "if there is some indirect force that connects us, we'll see each other again".
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I don't know if I wrote makes sense but this is what I gathered from reading through dictionary, encyclopedia, wiki and such. I'm sure someone more well versed in Buddhism can see the flaw in my logic or my choice of words. My brain is fried at this point.
