Of course the spell won't resolve, but is the indestructible creature a legal target for a spell with the effect "destroy target creature"?
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.
114.2. Only permanents are legal targets for spells and abilities, unless a spell or ability (a) specifies that it can target an object in another zone or a player, (b) targets an object that can’t exist on the battlefield, such as a spell or ability, or (c) targets a zone.
700.4. If a permanent is indestructible, rules and effects can’t destroy it. (See rule 701.6, “Destroy.”) Such permanents are not destroyed by lethal damage, and they ignore the lethal-damage state-based action (see rule 704.5g). Rules or effects may cause an indestructible permanent to be sacrificed, put into a graveyard, or exiled.
Unless it says
"destroy target creature that's not indestructible" than yes, you can target it.
And yes, the spell will resolve. Gideon won't be destroyed, but his controller will still get his
The same goes for spells that
cannot be countered and tapping and untapping creatures.`
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.
609.3. If an effect attempts to do something impossible, it does only as much as possible.
So in this case, the following happens:
Destroy target creature. It can't be regenerated.That creature's controller puts a 3/3 green Frog Lizard creature token onto the battlefield.
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'
If Rapid Hybridization resolves and the creature isn’t destroyed (perhaps because it’s indestructible), its controller will still get the Frog Lizard token.