Cappies are cool pre-Jihad for being the underdog. Not a big fan of them post-Jihad though, they've lost the underdog status, plus not a fan of Sun-Tzu's son not being a reformer like his father.
Their 'Mechs are good reason on nearly all eras though. In the Age of War, they created the Thunderbolt (ok, Earthwerks did and they're interstellar company BUT they did do it in the CapCon territory), their 31st century output is cool and innovative, and they got some nice toys during the Dark Age.
The FedSuns is another favorite, on the account of them often trying to be the good guys, though looking through their history, that hasn't gone so well often. For once, they could try something else than "let's beat the crap out of them", in a sense they're amusingly similar to the Clans.
Of course, the real reason is that many, many of my favorite 'Mechs are common in their hands, like the JagerMech. Not a fan of how they got some powerful toys from the Star League era though (such as the Thunder Hawk, Devastator or Nightstar).
I like the Free Worlds League in that they're actually nominally a democracy. Pity they couldn't curb the Captain-General's power. Not a fan of the resurrected FWL, doesn't feel the same anymore.
Unfortunately, 'Mech-wise the FWL get the short end of the stick, both in stats and looks, so i can't say i play the FWL... At least they got the coolest general purpose battle armor in the Inner Sphere in form of the Longinus.
The Republic of the Sphere is also pretty fun, in that they combine being the good guys with supertech taken from the WoB. Kinda interesting combination. However, since their demise at least to an extent is so inevitable, i never formed a real a connection to them. I don't mind my favorite factions losing or even getting destroyed completely, but the ROTS feels at times like it is made to be thrown away, though the few 3250 bits we've seen indicate something of them may survive.
Of the Clans, i like the Jade Falcons for their hidebound yet successful methods. Unfortunately, this has taken backseat to utter insanity during the Dark Age, and cool 'Mechs are barely a compensation for that.
Honestly nominating them for annihilation at this point.
Wasn't originally a fan of the Wolves but warmed up to them, especially during the Dark Age since it seems they've managed a reasonable balance between liberal attitude ("let's form an empire but not kill everyone" and 'Mechs like the Wulfen), and taking the seat of Crusader-minded top-Clan the Jade Falcons held previously. (Never took the Warden attitude quite seriously. It is like "look, we're good guys but we are beating the crap out of you first, no offense". OK, they didn't quite intend to do that but that's were they ended up.)
Finally, i like the Word of Blake for being so mustache-twirling villains.