Here are some jokes I made up (almost all are 'original' but a few are so common so it's just an adaptation of a well-known joke). I tried to keep the humour similar to the characters in the movies. Most are just context related puns.
Hope you like it
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Aquata to Arista: You're so shellfish
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Ariel: So while I was organizing my grotto...
Andrina: *rolls eyes* cave
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Ariel complaining about Sebastian: I just can't have any privacy in this fishbowl
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Ariel to Eric: No, I'm not going to wear fishnet stockings
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Alana: I look awful! Like something the catfish dragged in
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Attina complaining about taking care of her sisters: I'm drowning over here
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Aquata: *trying to dance* C'mon! Do the wave.
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Andrina: Attina's snoring could wake the Dead sea.
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Ariel to Flouder: You're such a chicken of the sea, aren't ya?
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Arista: *looking at a mermaid magazine* Check this out, she's all fin and bones.
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Alana buying new shells: Do you have anything in size C?
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Adella: Maybe I should try Plenty of Fish?
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Trition: Want to play water polo Ariel?
Ariel: Okay, I'll get my seahorse
not really funny, but makes 'under-the-sea' sense-
Alana: These scales always add a few pounds, agh.
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Trition playing the calliope: Hmm, this doesn't sound right. I think we need some tuna fish.
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Attina: Ah hah! Caught you red-handed, Ariel. Care to explain why you went to the Sea Witch - or - are you speechless?
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Ariel to Eric: You want to come to my world? Hold your breath.
Eric: Why? ... do I have to wait?
Ariel: No, my world is underwater.
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Triton: Ariel, you're obsession with humanity is disturbing. You need to sea kelp.