The country was and still is pretty centralized, so whoever controls the capital will probably end up controlling the rest in short order, especially because the army generally listens to who is in charge "in the center".
So, in order to get a civil war going, nobody has to effectively contorl the center (army HQ) for quite some time, enough for individual units to be more loyal to local authorities. Even then, I have my doubts.
That said, here are a couple of factions that could rise up:
1. Ceausescu loyalists
Pretty much everyone hated him, so they are likely the first to get taken out. Investing more in a guerrila-type force filled with fanatics to be used in case of war with the soviets and using that would help (limited efforts towards this end were apparently made), and so would bringing in foreign mercenaries (like everyone thought was happening in the general confusion of the day).
There might also be a case for creating a separate branch of the armed forces, kind of like the SS, Republican Guard etc. years before the revolution and then having that force stay loyal to Ceausescu.
2. The party
OTL, a most of the power structures of the state were taken over (once the army switched sides) by men (e.g. Iliescu) from the lower party echelons rebranding themselves as revolutionaries. They shared a lot of power with:
3. The Securitate
Just like in the case of the party, men form lower echelons of the Securitate came to lead various revolutionary comitees etc. OTL, between the party and the securitate there was a bitter rivalry for power. Have the revolution hijacked and split between these two competing forces.
4. The Hungarians
There was some real ethnic tensions in the early days. IIRC, one romanian policeman had his head cut off in a hungarian-majority region. To his credit, Iliescu did a lot to defuse tensions. In a much, much more volatile situation, things could get ugly, and, with troops withdrawn to fight in other places, hungarian regions in Transilvania (e.g. Harghita, Covasna) could obtain de-facto independence from Bucharest during the chaos with all sorts of militas running around.
5. The monarchists
When King Michael arrived in Bucharest a few years after the revolution for the first time, one million people from all over the country came to cheer him. Even in 1989, people were chanting his name in Bucharest squares before Iliescu grabbed the reigns of power. I see the area around Timisoara in the west supporting him if they feel the revolution in Bucharest has been hijacked, as well as a significant percentage (though probably not a plurality) of people with a higher education in the big cities. The church will also probably back him as well, thus giving him a huge edge in the rural countryside.
6. The Miners
In the chaotic years after 1989, the miners from the Jiu Valley had a huge influence. The communists tried to use them to restore order in Timisoara (where the revolution began), only for the miners to join the revolutionaries and tip the balance in their favour. Later, they would march on Bucharest, beating supporters of opposition parties who challenged Iliescu, students and even bringing down governments. They are a big wild card and whoever they side with gains a huge edge.
All of this assumes that the army command fractures, because, if it doesn't, there is no civil war.