Are brown dwarfs useful for gaining velocity via gravitational slingshot (or Oberth Maneuver, I'm still fuzzy on the difference to be honest) by a theoretical starship with fusion propulsion looking to gain velocity while conserving fuel? I have been informed, correctly or not, that a brown dwarf wouldn't have enough velocity for a ship to gain a significant boost in the turn, (would it make a difference in velocity if it were part of a binary pair?) but I wanted a second opinion.
It's kind of tricky to explain and it takes some visualization.
Any massive object can be used for a gravity assist and several planets have been used. The Moon (edit, Mars has been used, see first comment) has never been used cause it's too small. Venus and Earth, were both used for the Mercury mission and all 4 of the larger planets have been used. Gravity assists are a good way to leave the solar-system.
It's both the size of the object and the relative motion that make the assist possible. Jupiter, for example, would provide better gravity assists if it was closer to the sun and orbited faster.
The problem is, a planet or star can only provide a "push" in the direction it's going and stars like brown dwarfs tend to move in one direction. Think of throwing a ball into the path of a moving train, the train can only accelerate the ball in the direction it's going. Same with gravity assists.
Now you can direct the ball at a different angle, so the spacecraft's bend is anywhere from just above a 0 degree directional change to a full 180 degree turnaround, but it still only accelerates in the direction of the object giving the assist.
The 180 degree turn around provides the greatest acceleration (see picture)
But the obvious problem here with stellar travel is that you'd have to move in the opposite of your destination to gain that maximum velocity and given how far you'd have to travel to get to a star, that would be hugely counter productive.
More probable gravitational assists are usually less than 90 degrees in adjusted direction, where the increase in velocity is measurably less, but that's the tradeoff, you want to accelerate in the direction you want to get to.
Stars are so far apart and the "space velocity" (yes, that's the correct term), between them tends to be quite small compared to distance between them that it's not very convenient or likely to be of much help for interstellar travel.
Planets are far more convenient for gravitational assists because the solar system is all mostly on the same plane and because planets change their directional velocity a full 360 degrees every orbit, and even with planets, there's a window of opportunity, especially for the 4 outer planets, that NASA has to use to take advantage of gravity assists. The window resets about every orbit or two, which, if you consider Saturn's orbit of 29 years, there's a long wait between windows when Saturn can be used for a gravity assist towards Uranus or Neptune.
In theory, if there was a very fast moving brown dwarf star and it was moving in the right direction and relatively close, that could be used for a gravity assist, but currently there's no such stars in our vicinity and there might not be for hundreds of thousands, maybe millions of years.
A star with a large planet in a very low very fast orbit could be used but gravity assist velocity is still quite tiny compared to stellar distances. Gravity assists aren't likely to ever be all that beneficial to interstellar travel. Maybe if you were near the center of the galaxy with hyper-velocity stars, or, maybe when Andromeda and the Milky-way merge and you have stars moving in very different directions at significantly higher relative speeds, but generally speaking, yes it could be done, but it wouldn't be useful. It's probably far more practical to build faster spacecraft. Traveling interstellar distances with stellar gravity assists would be glacially slow. For near by stars, you'd probably lose time trying to use stellar gravity assists. For more distant travel you might find one that could give you a push, but at those speeds and those distances, your travel would still take tens if not hundreds of thousands of years. It's, in no way, practical.