I love Jesse Quick. She's smart, capable (runs her own business) and is a legacy hero in her own right from a separate line of speedsters (the Quicks) that often intersect with the Flashes. Her father Johnny Quick gained super speed by saying (visualizing) a "speed" formula and she inherited it. Her mother was the hero Liberty Bell.
So why should they intro Jesse Quick on the Flash TV show?
1) Well, if we're going to establish that there are other speedsters in the Flash TV universe, she's a perfect candidate that's not related to the Allen-Wests family.
2) She can come in, do her thing and then go off and live her life without becoming a regular (kinda like Huntress on Arrow).
3) Also, after the Wells/Thawne thing implodes, Barry is going to need someone to talk super speed with.
4) In those talks, she could be the philosophical foil for Barry. Thinking their powers are purely the result of (as yet unexplained) science gone wrong, to Barry's "the impossible" near-mystical speed force talk.
5) My least favorite reason, but one the writers will surely use because they like to drive a wedge between Barry and Iris, is they can use her as faux love interest like they did with Linda.
6) Lastly, if the Flash/Arrow spinoff does end up being called Justice Society, she'd be a great addition to that team with a lot of historical clout with the JSA (at least for comic fans).
So how to introduce her?
The easiest way is to make her a scientist working with Tina McGee over at Mercury Labs. Maybe her formula is the final piece to the tachyon problem they're working on? Tina and Jesse could have a parallel relationship like the one Wells and Barry have enjoyed only, you know, not evil and based on lies, which would mirror nicely with Tina's role on the original Flash show.
So before they bring on Wally or Bart, they should really consider bringing in Jesse Chambers. She'd be an interesting add to the show and when we inevitably have a big Flash family fight when the Reverse Flash returns (or maybe Savitar) they're going to need her.