I was today years old when I realized that low quality items have lower max durability and wasted Larzuk sockets quest reward on low quality flail. Since I have no chance of getting Honor Flail (Amn + El + Ith + Tir + Sol), is there any good alternative for Barbarian Frenzy that won’t require many sockets or maybe should I get another character and try to finish Larzuk mission?
So Larzuk always gives the max number of sockets a base can have based on it’s ilvl, regardless of if it’s low quality or not. It’s not clear from your post if you’re just annoyed about the low durability of an Honor you made, or if you didn’t receive 5 sockets, but low quality Flail with ilvl 41 (read on for more info about how to get ilvl 41+ for 5 sockets) would get 5 OS from Larzuk. So I think your issue was more where that flail dropped than what it’s quality was.
There’s a lot of great weapons for Frenzy. Honor is actually one of those that is good on paper, but as you’re finding, not easy to make due to the 5 OS requirements at the time when using it would be most awesome.
There’s a Horadric Cube recipe you may be able to use to get a 5 socket flail if you want to try again with a non-superior “normal” quality white item:
Ral + 
Amn + 
Perfect Amethyst + Non-superior white weapon → Socketed weapon of same type
Note that it will give you a random amount of sockets between 1 and 6, limited by the maximum sockets a weapon can have (base and ilvl restrictions). For a Flail base the max is 5 sockets, so you’d have 1/6 chance to roll 1, 2, 3, or 4 sockets, but 1/3 (2/6) chance to roll 5 because a roll of “6” will default to 5.
However, you also need a Flail with ilvl 41+ to be able to get 5 sockets, which means the only places in Normal difficulty a Flail with sufficient ilvl can drop are Worldstone Keep lvl 3 and Throne of Destruction, unless you’re in a game with a terrorized zone lvl 41 or higher.
Champion enemies are +2 area level though, and Unique and their minions are +3, so a Unique or minion in a zone lvl 38 and above could drop a Flail ilvl 41 (Abaddon, WSK lvl 1 & 2 in addition to what I listed for normal monsters) and a champion in a zone lvl 39 and above could as well (WSK lvl 1&2 in addition to what I listed for normal monsters).
That said Steel, Strength, and Malice are decent early weapons until you can get better. King’s Grace (scepters and swords) is also pretty nice at that point in the game. If you’re on ladder, Hustle is also a cool option, but needs a Ko rune which you likely wouldn’t have yet if you’re still in Normal.
Unbending Will, Oath, Lawbringer are also other good options when you get to higher level and have more runes available.
Uniques like Butcher’s Pupil Cleaver, Blood Letter Gladius, Hexfire Shamshir, Ginther’s Rift Dimensional Blade, and Headstriker Battle sword are also really solid through NM, but start to run out of steam at the end of NM and early Hell.
Dark Clan Crusher, Fleshrender, Sureshrill and Moonfall are also all unique exceptional grade maces that would likely be decent as well, since it seems like you were specializing in mace class weapons.
The unique Scepters count as maces as well in terms of weapon mastery, so they may be options as well, but be careful with RWs.