I feel a lack of explanation to why it was important to find Luke Skywalker. Sure Leia wants to, but Leia can’t waste their meager resources on a personal quest. What would motivate them better would be, however, if it was made clear that the main threat in the story is associated with the Dark Side and that they need someone trained in The Force to deal with it.

One thing I have seen proposed in review threads is that the Starkiller Base would be not a new construction - but an ancient lost Sith artifact.
That would not only provide motivation for finding Luke Skywalker but also explain how The First Order could have got something that massive operational in secret and why Kylo Ren (a dark-side Force user) needed to be on it.

I think that the idea of the Starkiller Base firing at Hosnian Prime after the Resistance has started attacking it is a great idea. Having the Rebels lose the main objective would break up the story being such a mirror of A New Hope.
Having the superweapon be a one-shot “doomsday weapon” that would destroy itself would also help explain why Han, Chewie and Finn don’t encounter that much resistance on it.
It being secret would make it more plausible why the New Republic has not mobilized against it.
If the Republic did not know about it / being operational, it would provide Finn with a stronger reason for defecting and reach the Resistance: To warn them.

Other things already on my personal list are:

Lots of people have complained about the Falcon taking too much of a beating when Rey flies it on Jakku.
When the Millenium Falcon is bouncing against the ground, add a shimmering effect to make it appear like there is a deflector shield protecting it against the worst hits.
Cut the shots of Finn using a lightsaber against Stormtrooper with lightsaber-deflecting baton. It should not be that easy for him to use it.
Cut or shorten the shot with the chasm appearing right between the beaten Kylo Ren and Rey. It is too convenient and it looks fake. (I heard that in the novelization, Rey chooses to spare Kylo Ren’s life because the dark side has tempted her to kill him. There may be an alternate take with that theme being stronger on the BR).
The ignition sound of the Skywalker lightsaber has been reported to be wrong.
Take the opening theme from another movie’s soundtrack. People think it sounds weak compared to The London Symphony Orchestra.
Make Han shout “Jacen” instead of “Ben”. 😉
Max Landis (accomplished script writer and vlogger) has proposed that Kylo Ren would be self-destructive after killing his father, and should taunt Finn and Rey in the duel by shouting: “Come on: Kill Me!”
(https://www.youtube.com/watch?v=TdEIzRUos4g)
Cut the whole bit with the Rathars… It seems to me like it was put there only to be able to make a reference to Indiana Jones.
Make the Starkiller fire one beam that hits one planet. Having the beams split up, each beam homing in on a different planet is just too implausible, over-the-top … It looks fanboyish. (That is, if it is going to fire at all)
Cut many of Domhmnal Gleason’s lines. He sounds like too much like a nazi, too much like a character in a fan-film.