Ludicolo is an amazing special sweeper in the rain, not having trouble with the bulky Water-types that populate OU, unlike most other special Swift Swim sweepers. Surf is your main STAB attack in the rain and a truly terrifying one at that, always OHKOing the standard 248 HP / min SpD Scizor with Stealth Rock in play and hitting almost everything that doesn't resist it for at least a 2HKO. A Grass-type STAB helps deal with most bulky Water-types; the choice of Grass Knot or Energy Ball comes down to how much trouble you have with Vaporeon. If you don't have much trouble with it, Grass Knot is preferred as it hits almost all other bulky Water-types harder than Energy Ball and manages a 2HKO on bulky Gyarados, even without Stealth Rock. However, if you have a lot of trouble with Vaporeon, then Energy Ball is preferred as it still hits most bulky Water-types hard enough, but unlike Grass Knot, it does enough damage to Vaporeon so that it can't stall you out by using Wish + Protect as Life Orb recoil slowly weakens you. Ice Beam hits the Dragon- and Grass-types that resist the Water- and Grass-type combination super effectively. It also hits Flying-types super effectively, but it must be noted that a neutral Surf in the rain is stronger than a 2x super effective Ice Beam.
The last slot is the most arguable when it comes to what to use. Rain Dance is usually the preferred option as you can never have too many Pokemon with Rain Dance on a Rain Dance team, and Ludicolo has the special bulk to set it up. Focus Punch is a great alternative, though, since with a Rash nature and 60 Atk EVs, Focus Punch + Surf in the rain and Stealth Rock damage will always take out 24 HP / 232 SpD Calm Blissey, relieving Ludicolo of what is usually the biggest problem this set faces. Other useful options for this slot are Hidden Power Electric and Hidden Power Bug. Hidden Power Electric always OHKOes 4 HP versions of Gyarados, and with Stealth Rock, the same goes for bulky versions, making it particularly useful if running Energy Ball rather than Grass Knot. Hidden Power Bug has a high chance to OHKO the standard 252 HP Celebi with Stealth Rock support, whereas Ice Beam can only muster a 2HKO. When running Hidden Power Bug, use 56 HP / 252 SpA / 200 Spe since the Speed drop from running Hidden Power Bug would otherwise mean you wouldn't outrun +1 neutral natured base 100s. Hydro Pump can be used over Surf since the extra power secures OHKOs on 252 HP Togekiss, 252 HP Metagross, specially defensive Forretress, and specially defensive Skarmory for example (all with Stealth Rock support). However, the accuracy loss makes it less appealing than Surf, which still has a lot of power in the rain.
If you're having trouble choosing between Grass Knot and Energy Ball, here are some damage calculations to help you:
Grass Knot vs. maximum HP Suicune: 79.2% - 94.1%
Grass Knot vs. 188 HP Vaporeon: 51.3% - 60.7%
Grass Knot vs. 156 HP Gyarados: 58.6% - 69.2%
Energy Ball vs. maximum HP Suicune: 63.9% - 75.7%
Energy Ball vs. 188 HP Vaporeon: 67.4% - 79.5%
Energy Ball vs. 156 HP Gyarados: 39.2% - 46.5%
A Modest nature and 196 Speed EVs reaches 225 Speed which means that in the rain, Ludicolo will outspeed neutral natured +1 base 100s. You could just max your Speed as this will at worst result in a Speed tie with neutral Breloom and Metagross outside the rain. When it is raining, however, you will outspeed 176 Speed EV +1 positive base 100s, but the extra bulk is usually more useful. Timid nature can be used since when it is raining, it allows Ludicolo to outspeed +1 positive base 100s and +1 +Speed Infernape, while outside of rain, Ludicolo will outspeed Jolly Tyranitar, neutral natured Mamoswine and Togekiss, and neutral natured Gyarados (or +2 Gyarados in rain). However, the loss of power will make it a lot harder for Ludicolo to deal with some of the bulkier Pokemon. Swift Swim is the obvious ability since this set aims to sweep in the rain where Swift Swim doubles its Speed. Life Orb is used since the extra power it provides helps against a multitude of Pokemon, getting OHKOs and 2HKOs it otherwise wouldn't get, but Damp Rock can be used if you have Rain Dance and want more Rain Dance turns.
Since this set aims to sweep in the rain, its teammates should either set up Rain Dance to support it or also try and sweep in the rain. Kingdra is a great partner to Ludicolo since the mixed set can 2HKO both Blissey and Snorlax (with a rain boosted Waterfall) who this set can't deal with if it lacks Focus Punch. Kingdra can also use Hidden Power Electric or Signal Beam in place of Rain Dance for Gyarados and Celebi, respectively, since both of them can trouble Ludicolo. Faster Kingdra sets can also aid Ludicolo by taking out some of the faster Choice Scarf users which trouble Ludicolo, such as Gengar. Kabutops and Qwilfish are both very effective partners to Ludicolo since they both OHKO Blissey and Snorlax with a +2 Waterfall in the rain. A few other things they can do to help Ludicolo are dealing with Celebi with X-Scissor and Poison Jab respectively, and hitting Tentacruel hard with Stone Edge in Kabutops' case or Exploding on it if need be in Qwilfish's case. Also, their secondary STAB attacks are very helpful for taking care of Abomasnow who walls Ludicolo to no end, unless Ludicolo can predict the switch in and Focus Punch. Qwilfish's high Speed also allows it to deal with Choice Scarf users such as Gengar, who Ludicolo can't outspeed, OHKOing it 100% of the time with a rain boosted Waterfall. Both Gorebyss and Omastar can make decent partners, especially if this set has Focus Punch, since although they are both slow, they have a lot more power than Ludicolo, and therefore with Blissey out of the way, they can proceed to tear into the opposing team.
The Rain Dance supporters who support Ludicolo best are simply those who can set it up most reliably and also get rid of Abomasnow and Tyranitar whose weather changing abilities disrupt Ludicolo. Hippowdon also changes the weather, but Ludicolo can use Surf or Grass Knot to kill it. Scizor is very effective since it has a STAB Bullet Punch to deal with the aforementioned Pokemon, can take care of Celebi with U-turn, and can also carry Brick Break or Superpower if need be for Blissey and Snorlax. Its many useful resistances allow it to set up Rain Dance for Ludicolo with relative ease. Although they are not common, Scizor is immune to the Poison-type moves that Ludicolo is weak to. Bronzong is probably the most reliable bulky Rain Dance set-upper there is so it makes a great partner to Ludicolo. It also has access to Explosion, which could help Ludicolo if Bronzong manages to Explode on one of the Pokemon who trouble Ludicolo. As a bonus, it also resists Ludicolo's Flying-type weakness and is immune to its Poison-type one, which may come in useful at times. Swampert is an effective partner since it is a great counter to Tyranitar and is also very reliable at setting up Rain Dance. It is also immune to Electric-type attacks, which while Ludicolo isn't weak to them, are still commonly aimed at it. Jirachi can also make an effective partner since it has many useful resistances and above average Speed, which allows it to set up Rain Dance quite easily and it also has a STAB Iron Head, which can take care of both Abomasnow and Tyranitar, assuming you get the 60% of a flinch. In the rain, Jirachi can also use 100% accurate Thunders with a 60% paralysis chance to slow down Choice Scarf users.