How do I get from Berlin to Abisko, Sweden? I have a Eurail pass, so I would like to make the most out of that.
I recommend to take train 94, the direct train from Stockholm to Abisko. November is the low season, and it's certainly the cheapest option for the last leg, if you are satisfied with just a seat. Even with a couchette, it can be quite cheap. If you book several months ahead for November, you can get an adult ticket for 550–650 SEK (65–75 EUR), or around 100 SEK more if you want a place to sleep, which I recommend. Youth or student tickets are cheaper, and if you're on Eurail you pay only reservation fees. The train also stops at Arlanda (there is normally an extra fee for boarding there). Train 94 is often late, so you might arrive Abisko only at 13:00, but that should not be a disaster. There is also a later sleeper train that departs 23:00-ish from Stockholm or Arlanda. I've taken this train many many times, far more enjoyable than a flight, on-board restaurant, nice views of forests, lakes, and mountains, and in November not crowded at all.
That leaves the travel from Berlin to Stockholm. There used to be a nightly CityNightLine to Copenhagen but this train unfortunately no longer exists. You will need to connect with various day trains, probably spending the night in Hamburg or Copenhagen. The good news is that the Swedish railways are good at guaranteeing domestic connections; if you have tickets to and from Stockholm, the train from Stockholm either waits for you, or you will be booked onto the next train. Just be sure to tell the train crew about your transfer.
In case of delays
For train travel within Europe, all connections are guaranteed in case of delays occurring in Sweden¹. I think I've travelled by train from Kiruna to southern Europe (Spain, Italy) five times now. Three times, a delay in Kiruna – Stockholm caused me to miss all connections. On every occassion, SJ replaced all connecting reservations without any bureaucracy. All I needed to do was to go to the customer service, show my Kiruna – Stockholm reservation, Stockholm – Spain/Italy reservations, and they would give me new reservations. Took less than 20 minutes, with no forms to fill and nothing to pay on my side. In summer, trains may get full, but this won't be a problem for you unless you travel around Christmas, so they can just give you the fastest connection as-is.
Ideally, I would like to get into Abisko in the afternoon, see the lights, and leave before midnight, get to Kiruna and take the train back to Stockholm. I don't even know if train times run that late, but the point is that I want to save money and time!
It is not possible to see the northern lights in Abisko without staying there for the night if you depend on public transportation.
In the low season (late September - early February), Abisko is served by only two trains per day. Exact times for southbound trains vary, but in some years the last southbound train departs 14:30, so you cannot see the lights. From the Luleå–Kiruna–Narvik timetable. You should really spend the night in Abisko to have a good chance to see the lights.
As an alternative, you could consider flying to Stockholm Arlanda Airport and take the train from there. SAS and SJ have an agreement to guarantee any connection flight+train (if you travel within Europe with SAS) and the sleeper trains from Stockholm to Abisko (either direct or with change in Boden) stop at Arlanda. This is a reasonable compromise between price, travel time, and a scenic train ride through Swedish forests, certainly now that sleeper trains (in particular international ones) are less and less common in Europe.
¹In theory, the same is true in other countries, but it can be much harder bureaucratically to get your right. For example, in Germany, I had to pay new tickets out of pocket and apply to get money back for the old ones that I could not use. That took months to be fixed.