Was there a plan to have Aragorn in The Hobbit?
Very early in the pre-production process, back when The Hobbit was only going to be a two-film series, Jackson talked about the possibility of showing Aragorn protecting the Shire in a "bridging" film set between The Hobbit and Fellowship of the Ring. In a 2006 interview with Ain't It Cool (emphasis mine):
Jackson: One of the problems with The Hobbit is that it is a fairly simple kids story, and doesn't really feel like The Lord of the Rings. Tonally I mean. It's always may be [sic] a little worried, but with two films that kinda gets easier. It allows for more complexity. At [sic] that implied stuff with Gandalf and the White Council and the return of Sauron could be fully explored.
That's what [the writers] talked about this morning. Taking The Hobbit and combining it with all that intigue [sic] about Sauron's rise, and the problems that has for Gandalf. It could be cool. That way, it starts feeling more like The Lord of the Rings and less like this kids book. You could even get into Gollum's sneaking into Mordor and Aragorn protecting The Shire. That's what we'd do. Love to work with Viggo again.
However, this statement was made long before Viggo Mortensen was asked about the role. In an interview with The Guardian, Mortensen discussed being asked (emphasis mine):
In Peter Jackson's Hobbit film, several of the Rings cast reprised their roles. Was he asked to take part? "No. Before they started shooting, back in 2008, one of the producers did ask if I would be interested.
In 2008, the script was still under development. In a 2008 interview with ComingSoon.net (emphasis mine) Guillermo del Toro, who was still attached to direct at this point, said:
"Literally, like every week, what you discover writing the two movies, writing the two stories, it changes. So, every week there’s a discovery, and anything we say this week would be contradicted next week.
It seems likely that Mortensen was asked not because there were any specific plans to put him in the films, but rather to give the writers an idea of what was or was not possible. Obviously it was not possible, and the films developed without him.
If he had accepted, what would have changed?
It's unknown how far Jackson would have gone to write Aragorn into the movies. Even if we assume they remained largely unchanged, Thranduil's comment to Legolas at the end of Battle of the Five Armies:
Thranduil: Go to the North. Find the Dúnedain. There is a young Ranger among them you should meet. His father, Arathorn, was a good man. His son might turn out to be a great one.
Suggests that some version of Aragorn (whether played by Mortensen or not) could have appeared in the films as they were.
maguirenumber6's answer, below, provides some interesting ideas on things Jackson could have done, had he written the trilogy to include the "bridging" film as originally intended, but we'll probably never know what he would have done.