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@Trapinch: Magnet Ball should get x3 catch rate instead of x1.5, since the Net Ball has that rate.

Health Ball
A Ball that works better on healthy Pokemon
x3 Catch rate if the Pokemon is not affected by any status.
(Not to confuse with Flarsable's Super/Stamina Ball)

Risk Ball
A Ball that is useful in desperate moments
x6 Catch rate, but if the capture is unsuccessful, the Pokemon faints.

Boost Ball
A tricky Ball that works better on Pokemon with boosted stats
+.5 Catch rate for each stat boost stage the Pokemon has.
Comments: Use Swagger/Flatter to catch Pokemon easily with this ball.

Swift Ball
A fast Ball that works better on slow Pokemon
x2 catch rate if Pokemon Speed is 150-100
x3 catch rate if Pokemon Speed is 100-50
x4 catch rate if Pokemon Speed is 50-0

Dizzy Ball
A strange Ball that works better on confused Pokemon
x3 Catch rate if Pokemon is confused

@Flarsable: I named it Swift Ball because the Ball is supposed to be thrown so fast that slow Pokemon wouldn't see it coming, reducing their resistance against the Ball. And a Ball with catch rate bonus of fast Pokemon would be unbalancing, as most high-level and Legendary Pokemon have high Speed.
@Sovereign: Gratitude Ball (PL) should be effective against Flying type, since Shaymin Sky Forme is Grass/Flying (according to Serebii.net)'

Hurt Ball
An spiked Ball that damages the Pokemon when thrown
x3 Catch rate. Each throw reduces the Pokemon HP by 1/8

Cloud Ball
A Ball that works better on Flying and Wind type Pokemon
x3 Catch rate if Pokemon is Flying or Wind type

Expert Ball
A Ball that gives extra EXP to Pokemon
x1 Catch Rate. Pokemon caught with this ball gain boosted EXP (same bonus as if they had been traded)

Golden Ball
A Ball that rewards the Pokemon with money
x1 Catch rate. Trainer gains money each time the Pokemon caught with this ball defeats a Pokemon. Money obtained is equal to (Pokemon level*20)

Well, I think is time to take Pokeball effects to the next level. Actually, maybe I just did that with Expert and Golden Ball because I introduced Pokeballs that give added bonuses to Pokemon, in the same fashion as Abilities (in fact, the effect of Golden Ball can potentially become an ability for a coin or money-based Pokemon)

Now, here are more "Advanced" Pokeballs:

Focus Ball
An stiff Ball that allows the Pokemon to endure an attack
x1 Catch rate. Allows the Pokemon to survive 1 fatal attack with 1 HP.

Miracle Ball
A marvelous Ball that protects Pokemon from ailments
x1 Catch rate. Pokemon cannot be afflicted by any non-volatile status ailment (Poison, Sleep, Paralysis, etc.)

Precise Ball
A Ball that boosts the Accuracy of the Pokemon
x1 Catch rate. Boosts Pokemon Accuracy 1 stage in battle (hidden value)
(I don't want to mess with other stat-boosting Pokeballs because you can only boost Accuracy in battle with just 1 move: Acupressure)

For keeping the game (or RPGs) balanced I suggest to limit these kind Balls to 1 per game/adventure. That means, these Balls shouldn't be sold and should be only obtained in the field, caves or maybe specials events such as Lottery.

And this is not actually advanced, but is a unique effect:

Random Ball
An odd Ball with a varying catch rate
When thrown, the Ball will randomly get x1, x3 or x5 Catch rate.

EDIT: btw, I noticed that all the names of official items, as moves, have a limit of 12 characters (including spaces) so I suggest everyone to give Pokeballs names shorter than 12 characters.