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.
The cheapest option Berlin – Abisko on EUrail
Berlin – Copenhagen, train 50473, 00:32 – 10:07
Copenhagen – Stockholm, train 536, 11:15 – 16:40
Stockholm – Abisko, train 94, 17:58 – 11:33
I recommend to take train 94, the direct train from Stockholm to Abisko. It departs at 17:58 and arrives 11:33 the next day. 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 now 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. Train 94 is often late, so you might arrive Abisko only at 13:00, but that should not be a disaster.
That leaves the travel from Berlin to Stockholm. You can either try to find a cheap flight, or take the train. The Berlin–Malmö train from Sergio's answer runs rarely if at all during the low season. To reach train 94 Stockholm – Abisko (see timetable here), you must arrive in Stockholm by the latest 17:38. In this case, your transfer in Stockholm is guaranteed: train 93 will either wait for confirmed passengers, or you will be rebooked to your next train. Just be sure to tell the train crew about your transfer.
From Copenhagen to Stockholm, train 536 runs every day except Saturdays. It departs Copenhagen 11:15 and arrives Stockholm 16:40. It's a fast train, but with a Eurail ticket you only pay the modest reservation fee.
The direct train from Berlin to Copenhagen departs Berlin Hbf nightly at 00:32, arriving Copenhagen 10:07. Warning, on some dates there are trackworks and the arrival in Copenhagen is too late. Again, reservation fees are modest on EUrail, certainly if you're satisfied with a seat.
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. In particular, the last southbound train departs 14:30, so you cannot see the lights. From the Luleå–Kiruna–Narvik timetable, northbound trains arrive 11:40 and 16:31, while southbound trains depart 12:00 and 14:30. 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 used to have an agreement to wait for each other through FlyRail, but I think that has finished, so your connection may not be guaranteed. Flight + train is still a popular compromise between cost and travel time.
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.