Early on, and for a long time, my main faction was the Word of Blake. This was because I was the one convincing my brother and my friends to play the game in the first place, and since they'd inevitably want to run a campaign with a unit of their own, I'd end up playing the opposing force in most games, and the Word of Blake was a popular opponent. Eventually I started to read up on the faction and in some ways I enjoy them precisely because they're nearly my polar opposite. But admittedly, part of what usually pulls me to a faction is that their units suit my play-style(s), and C3i plus iNarc (especially back in the days before ECM was all over the damn place) can be a pretty brutal combination once you really get the hang of them. I never played a campaign with them myself, but I still play Blakist opfors for Taharqa from time to time and I still always enjoy breaking 'em out.
Currently, my big factional attachment is Clan Hell's Horses. In the wake of the release of the 3145 material I was excited to play with all the new toys, so I set about putting together a new campaign as a side project from
my main campaign. Eventually, because I realized I wanted to do one each of Inner Sphere and Clan, I did both
a 3145 revival of House Ma-Tsu Kai (long my favorite Capellan unit) and
a Clan Hell's Horses Galaxy. Again, in both cases I worked backwards from unit selection befitting my play style, but as I've read up on both I've really grown to enjoy them, especially Clan Hell's Horses. (Note: Like the Word of Blake, my love of these factions paradoxically stems partly from the fact that I find them colorfully abhorrent.)
I've also always a big soft spot for the Free Worlds League, especially the Duchy of Oriente. They're probably closest to my preferences in terms of government (although to be clear they're still light-years off from it; suffice to say that BattleTech is not the setting I go to for that sort of thing), other than maybe the Rim Collection or the Outworlds Alliance. Also, at least in my opinion, they have more flavor than the other five Houses combined.