Back to the Future’s DeLorean with Marty on that first trip to 1955?
The most obvious change would be that Jennifer would have a presence throughout Back to the Future's first film and likely be instrumental in Marty and Doc's plans. She'd also consequently have a much larger role in Back to the Future 3, where she could have been especially instrumental to Marty's character arc.
Jennifer Would Help Marty Get His Parents Together
In this hypothetical, Jennifer would be with Marty when he goes to visit Doc. In the original Back to the Future, Marty learned that the family car was wrecked, so he wouldn't be able to take Jennifer on their date to the lake. This could lead to them deciding to sneak out late at night and see one another in order to make up for it, only for Marty to that he'd promised to see Doc early that morning. Jennifer would be with him, planning to part ways after the visit, thus getting swept up in everything when the terrorists arrive. She'd be forced to hop into the DeLorean with Marty, getting stranded with him in 1955, leaving them to work on finding a way back to the future.
With a version of Jennifer in Back to the Future who is also involved in its time travel shenanigans - taking her 1955 alongside Marty - she'd be able to help him get his parents together after his mother falls for him instead of his father. The easiest way to manage this would be for Jennifer to help influence Lorraine while Marty works on steering George. Jennifer could infiltrate Lorraine's friend group as a new girl and try to ensure that George is consistently brought up, just to keep him on Lorraine's mind. She could also provide Marty with intel on Lorraine's interests and give advice on how to coach George.
It wouldn't necessarily make every aspect of the ordeal any easier, or change the actual events of Back to the Future a great deal. Biff would still be a problem for Marty, after all, only causing Lorraine to fall even harder for Marty. However, Jennifer would simplify certain elements of the plan without question and add a whole new element to the first movie. It wouldn't be until Back to the Future 2, that the presence of Jennifer would have a massive impact, especially in the first and second stages of the film.
Jennifer Wouldn't Have Been Neuralyzed In Back To The Future 2
From the very beginning, Jennifer's role in Back to the Future 2 would be drastically different. Doc states that he brought her along because she'd already seen the time machine. He then put her to sleep because she was asking too many questions. However, this version of Back To The Future's Jennifer would have already been time-traveling with them, meaning Doc wouldn't see a need to leave her out, let alone put her to sleep, allowing her to be more of an active part of proceedings.
If Jennifer is never put to sleep by Doc Brown, then pretty much nothing that happens after the confrontation with Griff would be the same. Jennifer wouldn't be left in an alley by Doc and carted off to her 2015 home by the police. On the contrary, she'd want to have a part in the plan to save her future family from ruin. Not only that, but without Jennifer being put to sleep, Doc would be able to ensure that Marty Jr. would remain asleep for the entire hour, preventing the confrontation with Griff from escalating - or at least delaying it a little longer until Griff called Marty chicken.
Marty And Jennifer Would Insist On Seeing Their Own Future
Of course, if Jennifer isn't left asleep in the alley, then the entire second half of Back to the Future 2 wouldn't happen. Marty and Doc wouldn't have to go rescue Jennifer from the home of her future self, and Biff would never get a hold of the time machine and go back to give his past self the Almanac. Unless, of course, Jennifer proactively worked with Marty to convince Doc Brown that they should look into their own future.
Doc Brown lied to them, after all, when he told them that they turned out fine. A trip of this nature, just to check up on themselves, would prove it. This would potentially allow Marty and Jennifer to see their 2015 future for themselves, under the condition that all they do is look - which could easily set up a similar dynamic to the original movie's plot. While Marty and Jennifer busy unsuccessfully messing with things to make their futures less miserable, Biff could get a hold of the time machine and head back with the Almanac to set up the next half of the movie.
Jennifer Wouldn't Have Been Left In Biff's Alternate 1985
Another radical change would be in Jennifer's continued presence throughout the rest of the film. As a result of her experience in this type of scenario beginning with the first Back to the Future, Jennifer would be far more careful not to bump into her future self. As a result, she wouldn't go into shock and out. Without her being asleep, there'd be no reason to leave her in the alternate 1985 created by old Biff.
A far more likely scenario would be for her to discover the alternate 1985 with Marty, with the pair together learning about the new circumstances of this hellish new timeline. They'd inevitably learn of Biff's newfound fame and wealth, tracing the whole thing back to him after Doc discovers - on screen, this time - the head of future Biff's cane and the Almanac purchase bag in the DeLorean.
With Jennifer now serving an active role in all of this, she'd be with Doc Brown while Marty learns when young Biff got his hands on the Almanac. Therefore, she'd be around when Doc arrives to rescue Marty from Biff, and she'd them in 1955 once more. This time, Jennifer would be able to help them execute their plan to get the book back from Biff. With her help, Marty wouldn't get trapped in Biff's garage, so Doc would never bike out to Biff's house. Instead, he'd stay on standby with the DeLorean.
If the original events of Back to the Future 2 are anything to go by, Marty and Jennifer would struggle to get the Almanac away from Biff since he has it on him at all times. It'd be even more difficult to pull it off in broad daylight, without the cover of the dance or antics of the first Back to the Future to distract from their pick-pocketing efforts. They still wouldn't get a fair chance at it until the dance, but it might initially be far easier with both of them there, using either of them as a distraction while the other grabs the book.
Of course, Marty's character arc wouldn't be complete yet, so he'd likely still get baited by Biff, which would allow the final confrontation to still occur in the same fashion. This new version of Back to the Future 2 would end on a similar note to the original. However, there would likely be one significant change: Doc Brown would no longer be the only one whisked away by the lightning strike.
Jennifer Would Be Around In Back To The Future 3
As she'd also be in the DeLorean, Jennifer would get sent to 1885 alongside Doc Brown, giving Marty even more reason to head back for Back to the Future 3. Getting help from 1955's Doc Brown, Marty would head to 1885 to rescue Doc Brown and also his girlfriend, raising the stakes. As a result, Jennifer would have a presence in the third film beyond the final few scenes.
Jennifer's role in Back to the Future 3 would primarily help to reinforce Marty's character arc. Thanks to her involvement, Marty would know about his future self's termination earlier. He'd also have the recent experience with Biff still fresh in his mind. He'd still learn about his ill-fated ancestor from Seamus McFly, as well. Jennifer's presence as his biggest er and a confidant who's been through all of this adventure with him would provide a bit of perspective for Marty that'd be closer than what Seamus offered in the third film, especially as she'd been through it all first-hand, and wouldn't exactly be blameless herself.
Jennifer would've also been able to provide some amount of for Doc Brown's Clara Clayton romance as well. Doc spent eight months in 1885, based on his letter. With that much time alone with him, Jennifer would have had the opportunity to become closer friends with Doc as well. Then she could give him a bit of her own insight on the matter of his romance, possibly saving both him and Clara Clayton the heartache of Doc's goodbye, and sparing Clara from having to chase down a speeding locomotive.
In the end, on top of providing a little extra physical assistance, Jennifer would reinforce the emotional core of the series. She'd give Marty another peer to confide in and would help substantiate a lot of his character arc, through reflection on his actions, as well as actions of her own. The final lesson about one's future not being written could be applied to her just as easily as it was to Back To The Future protagonist Marty McFly, considering how invested she seemed to be in her future from what little presence she originally had in Back to the Future 2. Back to the Future is a trilogy that's all about making changes to established history, so it's interesting to think exactly how things might have gone differently if Jennifer had been given more to do.