Of course the spell won't resolve, but is the indestructible creature a legal target for a spell with the effect "destroy target creature"?
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Yes, Indestructible creatures are legal targets for spells/abilities that say "destroy target creature."
The spell actually does resolve, but when it attempts to destroy the creature rule 700.4 prevents it from happening. The requirements for a "legal target" for a spell must be a permanent, unless the spell/ability specifies otherwise. Indestructibility does not modify the rules for targeting, unlike the rules for protection (702.15b A permanent or player with protection can’t be targeted by spells with the stated quality and can’t be targeted by abilities from a source with the stated quality).
You can target Gideon with Rapid Hybridization, and when it resolves it will fail to destroy Gideon and will create 3/3 frog owned by Gideon's controller.
Unless it says
And yes, the spell will resolve. Gideon won't be destroyed, but his controller will still get his
The same goes for spells that
You can target anything that matches the description given for what's a legal target (unless Shroud, Hexproof or similar prevents you). Rapid Hybridization has you pick a target creature. Since Gideon, Champion of Justice is a creature (at this time), you can pick it as the target.
But Gideon is indestructible (at this time). It can't be destroyed [CR 700.4], but Rapid Hybridization instructs you to destroy it. Can't trumps do [CR 101.2], so you're asked to do something impossible.
So in this case, the following happens:
Gideon's controller will get to keep Gideon, and he will gain a Frog Lizard.
this is from the ruling section from the gatherer link for 'Rapid Hybridization'