An earlier question asked whether there were any planned missions outside the solar system, which made me think of this question. What would the scientific benefits be in sending a probe outside the solar system? By this I do not mean to other systems, but to explore near the outside of the solar system.
This depends very much on how extrasolar the mission is.
With current technology, we can just about send a probe to the edge of our solar system. This allows us to measure the interstellar medium. We've done this with the Voyagers.
With enough investment, we could send a probe to the Oort cloud and see what it's made of.
With future technology, we may be able to visit another nearby star, and examine its planetary system. That would broaden our understanding of how stars and planets are formed. It would also enable us to observe distant stars from two vantage points, enabling things like parallax measurements we can't do now.
With more exotic technology, we could travel to distant phenomena like black holes and pulsars and observe them up close. We could look at the sector of sky that's currently obscured by the Milky Way, and see what the Great Attractor consists of.
And that's before we've found habitable planets or even life.
First, start with the question "how do I convince an organization like the U.S. Congress to allocate funds for the mission?" In other words, "what tangible benefit might I receive from this mission or a future mission for which this one will prepare us?"
When talking about interstellar missions, the answer is simple: some day we may need to move to a new home. Where will we go? How will we get there? What would the journey entail?
Before sending humans anywhere, it is cheaper and more effective to send probes so we can build a better manned spacecraft. What questions might we need answered before we can send humans out of the solar system?
- What, exactly, is the interstellar medium made of and how dense is it?
- How much ionizing radiation is there outside of the Sun's sphere of influence? Does the Sun have a bow shock? If so, what changes after the boundary? If not, is there some region where harmful radiation changes?
- How difficult is it to communicate between Earth and a spacecraft in deep space?
- Is [insert propulsion technology X] viable for propelling a spacecraft to another solar system in a timely manner? Note that "timely" may still be many decades or centuries, e.g. using a generation ship.
We can always perform calculations and create models: however, we can likely get better results through direct measurements. Perhaps our models are wrong: direct observation may allow us to create more accurate models. This is how science works. All of the ideas I presented above involve models that we should refine and prove to be accurate before we send humans out to the stars. Otherwise, a manned journey may end poorly for those first extra-solar astronauts.