Things that can ruin Rhydon's Day
Tangrowth
Cofagrigus
Lilligant
Blizzspam
As far as I can see, we need 2 partners that can stop 2 of these things.
Cofag can be stopped with a Fighting-type and so can Blizzspam.
That leaves Lilligant and Tangrowth.
To damage Tangrowth the most, it must be a special attacker.
To damage Lilligant, it must be faster and it must hit hard. A physical attacker is better on Lilligant because of the SpDef boost, but she's pretty fragile, so I don't think it matters that much.
Both of these are Grass-type.
Knowing this, I think the two partners should be A Guts Fighting-Type (Haryama) or Munchlax (Thick Fat + RestTalk. He can also absorb some Sleep Powders if he needs to) and either Moltres (Lilligant and Moltres have same Base Speed), Galvantula (Faster than Lilligant), Accelgor (Really fast, Base 100 Special Attack), Sawsbuck (Counters both with Sap Sipper), Bouffalant (Sap Sipper), or Porygon-Z (Ice beams from this bro are going to hurt, has same Base Speed as Lilligant)