How to obtain a specific plant for the Zen Garden?

I'll assume you are talking about plants that you find for your zen garden (the only other option is the plants unlocked while playing the game's adventure mode, and those have a set order which never changes).

There is no way to increase the probability of receiving a certain kind of plant for your zen garden. The type of plant received is completely random.

Looking around the internet, it is said that while there is an equal chance to get all types of plants on Survival: Endless, other minigames have a higher tendency to produce plants you first unlock in the area where the minigame is played (night levels will produce more night-time plants, for example). This is not backed up by any hard evidence, however, and I can't really say I've noticed this during the countless hours my wife strived to acquire every possible plant for her garden.

I believe the game hints mention that certain types of plants only appear in certain types of boards or levels.

For aquatic plants, you need to be playing an aquatic level.

For plants for your fungus garden, you need to be playing a night level.