For now, but I promise I will add more if they come to my mind, here are some of the best… or worst, depends on the point of view:

There are three of a kind:

À plus dans l’bus
See you later in the bus

À bientôt dans le métro
See you later in the subway

À demain dans le train
See you tomorrow in the train

Note that l’bus, in the first one, is a contraction to keep the rhythm of the sentence but that is not grammatically correct.

^{Source for those: https://www.elle.fr/Love-Sexe/Mon-mec-et-moi/Articles/Les-expressions-tue-l-amour-2701522}

This one doesn’t rhyme but is silly enough to be here:

À demain, à deux pieds
See you tomorrow, on two feet

This is just a pun of language and cannot really be translated, it is a pun based on the fact that demain sounds exactly as deux mains; so two hands. Two hands, two feet; the pun is just based on this.

Another one from the same barrel:

Allô, à l’huile
Hello, in oil

Also just a pun of language based, again on the fact that allô (a greeting you would only use on the phone with someone), sounds exactly like à l’eau, so in the water… Once again, water, oil are two liquids, so they are associated in the pun.

^{Source for those two: https://www.lefigaro.fr/langue-francaise/actu-des-mots/2016/12/28/37002-20161228ARTFIG00002-dix-expressions-a-eradiquer-en-2017.php}

One that does not rhyme, but has the same amount of syllables in the two part of the sentence:

Si on se voit plus, on met des lunettes
If we don’t see each other anymore, let’s put some glasses on

I have to confess that I have no clue where this one is coming from but seems to be a pun based on the fact that words to say goodbye like au revoir, are implying that we have to be able to see in order to see each other ever again, and that, if we don’t, then, maybe it is just that we don’t see clear enough anymore.