I'll try to answer to some of your questions.
What makes structures so interesting?
Why give us the study of structures so powerful tools?
One very common and useful thing in mathematics is abstraction, a.k.a. generalization. Thinking in the language of structures allows mathematicians to generalize many properties to lot of different objects: for instance Cauchy's theorem tells us that for every prime divisor of the cardinality of a finite group there is an element that has order that prime, no matter if it is a group of permutations or isometries or matrixes.
Basically it is the maths version of the "two birds one stone"...although "lots of birds one stone" would be more appropriate, at least in my opinion.
Another important thing given by the notion of structure is the notion of (homo)morphism. Via morphisms we are able to relate and confront different objects having similar structure and this is useful for discoverying and proving properties of different objects: for instance in group theory we can study properties of different groups through their homomorphism in groups of permutation or groups of matrices (there is an entire field in maths that deals with these objects called representation theory, it has important applications outside mathematics for instance in physics).
Note that without the notion of structure the notion of morphism cannot be stated.
We could go on for very long answering just these two questions but I'd rather stop here to avoid being too boring (feel free to ask in the comments for addional stuff).
What is the motivation behind the definition of the term structures? (For example, why can't structure have more than one carrier set?)
Probably you should be more specific on which notion of structure you are referring to because there are multi-sorted structures which have families of sets as carrier. Examples of these multi-sorted structures are modules over generic rings (or if you like a more specific example vector spaces over different fields). Another example that I cite (because I'm a little biased) is that of categoy, which is a multi-sorted structure.
How did the term structure historically develop? What were the first motivating examples?
I'm not sure about that but I think that it came first during the 1800 when english algebraists observed the similar....structure undelying different kind of algebras and that through that notion they could prove a lot of intersting properties (like existace of solutions to equations) for large class of structures at once by reasoning in term of these abstract structures, instead of redoing the same work for all the concrete examples. These observations led to what is called modern or abstract algebra.
Why did one choose the name "structure"? I personally interpret this word ("structure") as being some kind of pattern. What has this to do with the mathematical notion of structures?
This question is itself your answer. The various notion of structures capture different common pattern in large class of objects. For instace the notion of group capture a pattern common to symmetries, isometries and various kind of number-systems: namely the fact of having a binary operation that is associative, with unit and inverses. Similar arguments apply to other kind of structures (rings, vector spaces, etc).
Hope this (maybe too long) answer may help.