Jigglypuff is debatably the best doubles character in melee, so she can pretty much partner with anybody. However, the most popular options are definitely the spacies.
To quote Armada, "Fox or Falco.
Fox is overall super great in teams and is very good at every aspect of the game. Wave shines into rest is super good vs floatys.
Falcos shines is a ****ing stupid move into rest. On reaction you basically will get tons of free rests each match. The Falco need to be aggresive and good at create good situations though but still. Falco is also very good at wake her up after a rest (even across the entire stage or even OFF STAGE).
Fox/Falco is the two best options for Puff in teams if you ask me.
The biggest reason for it is basically that Puffs team8 should be able to create a quick team combo (one hit then the Puff makes a rest right away). Fox/Falco play in their own league when it comes to being able to pressure the opponent.
They also have a super easy time getting grabs (shine grabs is stupid)
Have moves even from 0 that works pretty good as team combo starters (especially Falco on this one but still)."