God favorite weapon, I'm assuming just based on what I like as opposed to what I consider best... I have to break this up.
Guns:
A light shining in the darkness. I love this thing. Tiny powerhouse of speed and ferocity. Just clears any enemy that isn't just a meat wall. Deathclaws? Not an alpha or mother? No problem.
The Lucky. While it's still just a .375, it's crit chance on a crit build really makes it compete with other guns. Though, in all reality a crit build with like That Gun, or any other revolver and pistol would still out damage it, it's still just a beautiful gun.
Ranger Sequoia, while I believe Lucky is better looking, my first playthrough I was obsessed with getting and using one. I waited and killed fiends for like days of real life time to get ammo for it.
Moving on from pistols, I'm honestly fond of the lever action shotgun. Still outclassed by the hunting and Riot Shotgun (But the riot really outclasses basically most guns in the game... or weapons really.) it has some nostalgia, and lever actions just tickle me right.
As for rifles I really do enjoy the Bozar. It's somewhat inaccurate, not too high damaging, but it's scope and rate of fire sure make it fun.
Energy Weapons:
The MF Hyperbreeder Alpha is probably my favorite energy weapon. It feels nice to me to not have to reload. Just, wait a few moments and keep firing. Feels more natural. I feel stuck when I'm reloading, but not a problem with this. As well it does good damage, good speed, overall... for what it does it's just a good carrying weapon. Won't take down big threats...
The Beaten Protoype Tesla Cannon. I loved getting this, matched my gannon family tesla armor, and deal great damage. I'm sad the AOE was taken away, but multiple shots, higher damage output now I believe, and cool looks really makes it a nice high damage energy weapon.
Cleansing fire. You doing a flamethrower build bent on killing everyone you can in the game to get basically the most dull ending ever of "Everyone died in this town... nothing happens tbh." This is for you. Blue flames... What is this Avatar. Am I Azula? AM I BURNING DOWN NOVAC!?!
Elijah's Advanced LAER: While I find it... boring almost... I've just used it so much. Just... modded it has so much damage output it's crazy. Sure it breaks down like it was made from the hopes and dreams of Greek stock investors... but damn it's fine.
Melee/Unarmed:
Paladin Toaster: It just looks cool. It's not the best, but it's good against robits and power armor... but you look damn fine wearing it.
Salt Upon Wounds Fist: It's fairly simple, a powerfist covered in salt. However, it's flavorful. Who wield's it? Salt Upon Wounds? Where'd he get his name? This Fist? Why does he have such a cool weapon and helmet? WHy are you punching me? Why am I covered in salt now!?! WHO KNOWS!?!
Love and Hate: It's just cool. It's interesting to get, has a nice look. Overall rather dull, but I want to point out that... they put it into the game. Even a dull concept deserves to be put in. It adds flavor from across the spectrum, and is a great example of who New Vegas weapons are great. There's a unique for everything, and the uniques are... Unique, not just some randomly slightly better weapon... -cough- fallout 4 and 3 -cough-
I really only want to bring up one melee weapon really, the Blade of the West. You like legate lanius? Yeah? You like his sword? Yeah? Well, take this. It's better and is awesome for it. It's almost easy mode New Vegas. Crit chance to knock someone down. + Your 30% crit rate from that one perk... Legendary Deathclaw? Where? I knocked it down and kept it down as I hit it with a sword all day. Come at me New Vegas. I don't like it because it's broken... but it's awesome... being broken is just a +.
Explosive:
MFC Grenades: Easy to make, super cheap, lightweight, and good damaging. I don't really bother with other grenades. To me grenades don't really have to damage much, just damage enough. Sure like a Nuka grenade does more, but I like saving those as they are hard to make. These I can spam and spam maliciously.
25mm Grenade Launcher: I love this thing to rain down explosive shells like it's World War I. It's practical for most purposes. I won't really hurt myself. I like it unmodded from the extended barrel to have a nice arc. Just feels nice and practical.
Mercy: You want to become death, destroyer of worlds? Probably killing yourself in the process. Take the 40mm GRENADE MACHINEGUN. It... just deals raw unrelenting damage. Sure gotta get through the legendary deathclaw... but you earned it bub.
Finally, the Red Glare. A scoped, automatic, rocket launcher with cool red explosions, a lot of mods, easy to get, a lot of ammo available, multiple ammo types... It's just pure fun. You feel so versatile. It's not like the rocket launcher with it's lengthy reload. Or the Mercy which is a pain to get. It doesn't feel like you're cheating with power, just being rather good for most situations. Plus it looks... Really darn good and has a great reload animation.