What does it mean to hydrate an object?
Hydration refers to the process of filling an object with data. An object which has not yet been hydrated has been instantiated and represents an entity that does have data, but the data has not yet been loaded into the object. This is something that is done for performance reasons.
Additionally, the term hydration is used when discussing plans for loading data from databases or other data sources. Here are some examples:
You could say that an object is partially hydrated when you have only loaded some of the fields into it, but not all of them. This can be done because those other fields are not necessary for your current operations. So there's no reason to waste bandwidth and CPU cycles loading, transferring, and setting this data when it's not going to be used.
Additionally, there are some ORM's, such as Doctrine, which do not hydrate objects when they are instantiated, but only when the data is accessed in that object. This is one method that helps to not load data which is not going to be used.
With respect to the more generic term hydrate
Hydrating an object is taking an object that exists in memory, that doesn't yet contain any domain data ("real" data), and then populating it with domain data (such as from a database, from the network, or from a file system).
From Erick Robertson's comments on this answer:
deserialization == instantiation + hydration
If you don't need to worry about blistering performance, and you aren't debugging performance optimizations that are in the internals of a data access API, then you probably don't need to deal with hydration explicitly. You would typically use deserialization instead so you can write less code. Some data access APIs don't give you this option, and in those cases you'd also have to explicitly call the hydration step yourself.
For a bit more detail on the concept of Hydration, see Erick Robertson's answer on this same question.
With respect to the Java project called hydrate
You asked about this framework specifically, so I looked into it.
As best as I can tell, I don't think this project used the word "hydrate" in a very generic sense. I see its use in the title as an approximate synonym for "serialization". As explained above, this usage isn't entirely accurate:
See: http://en.wikipedia.org/wiki/Serialization
translating data structures or object state into a format that can be stored [...] and reconstructed later in the same or another computer environment.
I can't find the reason behind their name directly on the Hydrate FAQ, but I got clues to their intention. I think they picked the name "Hydrate" because the purpose of the library is similar to the popular sound-alike Hibernate framework, but it was designed with the exact opposite workflow in mind.
Most ORMs, Hibernate included, take an in-memory object-model oriented approach, with the database taking second consideration. The Hydrate library instead takes a database-schema oriented approach, preserving your relational data structures and letting your program work on top of them more cleanly.
Metaphorically speaking, still with respect to this library's name: Hydrate is like "making something ready to use" (like re-hydrating Dried Foods). It is a metaphorical opposite of Hibernate, which is more like "putting something away for the winter" (like Animal Hibernation).
The decision to name the library Hydrate, as far as I can tell, was not concerned with the generic computer programming term "hydrate".
When using the generic computer programming term "hydrate", performance optimizations are usually the motivation (or debugging existing optimizations). Even if the library supports granular control over when and how objects are populated with data, the timing and performance don't seem to be the primary motivation for the name or the library's functionality. The library seems more concerned with enabling end-to-end mapping and schema-preservation.