I'm not sure certain.
Cost is, at its core, based on physics- energy, resources and time are the true values all money is based on. Those constraints will exist everywhere in the universe, however they may be treated by an ETI. No matter where you go or who you are, you still need a massive amount of energy to move matter from one star to another.
The real problems are political, technological and ecological stability.
No one has ever built electronics capable of lasting hundreds of thousands of years. It's not obvious if it's possible because on those timescales everything crumbles to dust, especially if it's constantly being abraded by dust and radiation.
And if you don't put your travellers into long-term suspension, political and cultural stability are even more fragile.
And small eco-systems are far less resilient than planet-sized systems - which, it turns out, may not be resilient at all on longer timescales.
So the gap between launching Voyager I/II and launching a self-replicating probe which can reproduce itself physically and culturally is beyond huge.
If you happen to have massive amounts of energy some of these problems get easier, because you can get where you're going more quickly. But some remain immensely difficult.
Which is why percolation is a much better model. There will inevitably be some attrition in any colonisation program, and there may well be a hard limit on the amount of entropy any colonisation program can survive.
Might as well ask Archimedes to send a probe.
Hell, I'm not even sure we can make a computer that actually lasts 100 years.
I'm pretty sure we can. I have a 40-year-old calculator that is still working just fine.
Still, I was probably too ambitious when I said "computer". I was thinking more that we can't make a PC or server-style computer that would actually last 100 years, while in regular operation.
Also, 1980s calculators have much, much fewer moving parts than today's computers. They don't have any kind of active cooling, they don't go through rapid heating and cooling cycles, they probably have three orders of magnitude less electrical components than a modern PC.
For me at least, all of the PCs I have had so far have had a few electrical components giving out after at most 10 years (usually some condensers leaked).
In fact, it would not surprise me a bit if you could run a Raspberry Pi Zero for 100 years.
I wouldn't even be convinced then that we have the technology today. Any repair system would necessarily suffer from the same kinds of failures.
It's not even obvious that it is theoretically possible with current circuit technologies. Pieces would fuse together, moving parts would erode, all sorts of organic materials might deteriorate, everything would be affected by the huge amount of time. And you would almost certainly need energy to actively fight this entropy so that you would have an active machine when you reach your destination.
Iron is nuclear ash, and this tech is just about the most reliable implementation of logic we have. It was used in the V2 rocket and US nuclear submarines. It's abandoned tech though and was never pushed beyond 1MHz. If however you have a decade to compute your reactions then that might not matter.
Superconducting magnetic loops also store their flux indefinitely, and in space keeping it cool would be trivial.
I'm not sure any of these assumptions are applicable given billions of possible origins of life at various stages of development. Even at Earth, and with extremely conservative estimates, given 100 more years of nanotech/biotech/AI development, we won't die off, and could certainly send autonomous robot probes. When you don't die anymore, 8 years roundtrip (or more) suddenly doesn't seem so long.
BTW in the fiction arena, I suggest reading David Brin "Existence" if you're so inclined which is related to this :)