Lugia: Lugia is seen communicating with Ash telepathically in the second Pokemon movie. This could serve as evidence for other psychic powers. Also, Lugia is the master of the seas, which are affected by tides which are controlled by forces from the moon. This is also kind of telepathic. Overall, however, I feel like a Water/Flying typing makes much more sense, but there is some explanation for Psychic/Flying.
Exeggcute: Exeggcute heads originate from Exeggutor, which is some kind of plant, constituting to the Grass typing. As for Psychic, its heads communicate telepathically with each other, as seen in its Pokedex entry for Silver:
If a head drops off, it emits a telepathic call in search of others to form an EXEGGCUTE cluster.
Azurill: Its Pokedex entries say nothing about it having any special powers having to do with Water. As such, it is normal and thus has a Normal typing. However, it would also make sense to have a Water typing, seeing as its name is literally a combination of "Azure", which is a "blue color", and "Rill", which is a "small brook or river", and it is also described as living next to water, as seen in its Pokedex entry for Diamond:
A Pokémon that lives by water. It moves quickly on land by bouncing on its big tail.
Jirachi: Jirachi is associated with making wishes come true, as seen in its Pokedex entry for FireRed and LeafGreen:
It is said to make any wish come true. It is awake for only seven days out of a thousand years.
People often make wishes on shooting stars. Shooting stars, which are actually meteoroids hurtling through our atmosphere, often contain metal or metal alloys. Another explanation can be found in its Emerald Pokedex entry, which states it forms a tough outer shell, probably metallic, to protect itself while it sleeps:
JIRACHI is said to make wishes come true. While it sleeps, a tough crystalline shell envelops the body to protect it from enemies.
Celebi: Celebi is a Grass type. I don't blame you though... GameFreak making sense would be enough to confuse anyone.
Inkay: Here is Inkay's Pokedex entry for X:
Opponents who stare at the flashing of the light-emitting spots on its body become dazed and lose their will to fight.
Not only would flashing these brain-altering lights take psychic powers (hence the Psychic typing), it also inflicts harm with these lights (hence the Dark typing). However, it would make sense for Inkay to have a Water typing, seeing as it has the whole squid thing going on. I guess GameFreak decided Psychic/Dark was a better fit.
Gyarados: Apparently, there is a Chinese legend where a wimpy fish jumps up a waterfall and is granted the ability to fly. In context of Pokemon, Magikarp (the wimpy fish) must gain twenty levels as a wimpy fish, and is awarded with a strong, early evolution in Gyarados (the ability to fly). Here is the link to the whole tale, if you are curious. This also makes sense because Magikarp is described as being able to use Splash to jump over a mountain, which is similar to jumping a waterfall, in its Platinum Pokedex entry:
A MAGIKARP living for many years can leap a mountain using Splash. The move remains useless, though.
There are other theories having to do with problems that would have arised if Gyrados had actually been typed as Water/Dragon. Some say that getting a Dragon type Pokemon so early in the game would be OP, while others say that Gyarados would have had no weaknesses in Gen 1 (the only thing that hit Water/Dragon super-effectively at the time was Dragon, and the only Dragon type attack was Dragon Rage, which does a fixed amount of damage), and this would have been too OP. And there are also people that say that there were programming errors between Pokemon Red and Blue which made it impossible to type Gyrados as a Water/Dagon.
Helioptile: Helioptile is based on a reptile that lives in the sun, as shown in its X Pokedex entry:
They make their home in deserts. They can generate their energy from basking in the sun, so eating food is not a requirement.
While there is something that makes Helioptile a little extraordinary which constitutes to its Electric typing (it has frills which absorb solar energy and turn it into electricity, according to the Y entry) it is still based off of a reptile which basks in the sun and retains much of those characteristics. As such, it has a normalcy about it and therefore has a Normal typing.
