Lyn's popularity and importance in the West is based on her being the first FE lord for western gamers. I'd like to hear some other better reasons though.
PS: I'm not saying that it's not enough reason, btw.
Reasons...
Well, among the FE cast, she stands at 5th in relevance and importance, just behind Roy. Though unimportant and irrelevant to Japanese fans of the series. She's iconic to the western fanbase. Also, she was the only among the FE7 cast to be playable in the FE13 DLC, and is one of the few female lords in a series filled with blue haired male protagonists. Unlike Michaia, Lyn is generally more well-liked because she's not useless in game. Quite the opposite actually, landing criticals with Lyn (which happened rather frequently) was always a joy.
In retrospect, her high time was during Brawl days, this time around her chances are lower. However, she was an Assist Trophy and the 3rd FE character in Brawl, where Roy was missing (planned or not), so she still has somewhat of a shot, albeit a slim one.
Other reasons, being a female character helps diversify the cast, and is a sword user with a Japenese sword style, something we don't have at all in Smash atm.
The primary reason would be playstyle however. Lyn was a character I picked from the ground up simply because of the possibilities she brings to have a truly unique character that has a lot of competitive value. As someone who likes to play other fighters, I give a lot of merit to characters that are created with competitive play in mind, simply because it makes for a more well-thought out, balanced and fun character than "I want him/her in because I like the game and I feel s/he's important to Nintendo." Really, in contrast to other FE lords, Lyn is overshadowed by them in importance.
From my point of view she ties with Roy for relevance in regard to Nintendo, so take that as you will. However, also keep in mind that Roy was primarily added to promote his upcoming game, not because he was important to FE or Nintendo. This both helps and hinders Lyn at the same time as she's in the same boat. However, as I said before, she was made an Assist Trophy out of sheer demand for her character in Brawl days. So Sakurai is indeed aware of her popularity as are IntSys by making her DLC. If only Sakurai built Smash with a Competitive playstyle in mind, Lyn would have a MUCH higher shot at getting in.