I am aware that the pilot was leaked online and was met with mixed reviews, but I have not seen it and thus my opinions are based purely on trailers, posters and other rumors etc.
1. Focus On The Superheroics
From the trailers the show has given off a vibe of action adventure and not romantic drama like say Smallville was. The one thing I think most live action DC shows (mainly CW Shows) have been really bad at is romantic subplots. This show has a golden opportunity to truly be different than the CW shows in terms of storytelling. I am sure there will be romantic drama but if they go that route, at very least I ask that they make it convincing/ genuine and not forced or tacked on like say Oliver Queen and Felicity has become on the show "Arrow".
2. Supergirl Not Superman
I really want this show to stick to KARA and not keep making nods and slight reminders that she is the cousin of Superman. She can stand on her own, no need to slap us in the face that Clark Kent is related to her every chance given.
3. Feminism Done Right
This series will be aimed a women and little girls but males such as myself will be watching also. From the trailer there is a line that a villain says to Supergirl and goes something like this "On my planet females bow to males" and Kara responds by saying "This is not your world" indicating that the show wants to prove that women are as powerful as males. There are two types of ways of doing feminism, you have those that are trying to paint males as pig jerks with no heart or compassion and see women as lesser or the weaker gender.
That way is the wrong way to approach it for sure, then you've got the correct way to present it. Creating a strong female character that sees herself as an equal to men and not above males or hate all males based on previous ingrained negative notions. I believe little girls in particular will gravitate towards that and keep them coming back for more.
4. Memorable Villains
Whats a hero without a great villain? A sad hero. One thing the Flash series got right was the villains in particular the reverse flash. I am hoping that Supergirl does not follow "villain of the week" formula and does it's best to give Supergirl challenges that will question her place in the world and her heroic quest in general. Honestly it's not much to ask but the ball is in their court and I can only hope for the best.