This is certainly an important question. Here are some of my own thoughts on it:
Having the regular homework consist of questions that the students do, in principle, have access to on the web is not necessarily problematic.
For example, I recently finished teaching a year-long course out of Michael Spivak's acclaimed Calculus text. At the beginning of the year I found that I was able to freely download the solutions manual. I told the students that the solutions manual was available online and that it was their job to look at it from sparingly to never. In fact they did a good job of this: the amount of time that they spent coming to me instead, sometimes more than once per problem, and the amount of effort and verbiage that most of them put into their homework strongly suggests that they hardly ever consulted the answer book. (I remember one instance in which a student freely admitted that she had cited the answer book for a single problem, which I found most impressive.) It also helps that the answer book is on the terse side and the grader was very picky and detail-oriented. Finally, the grades they were receiving on their homework were not such a large determining factor in their course grade so as to tempt students to cheat. This I think is an important point in undergraduate classes: make the homework be worth a not completely negligible percentage of the course grade -- say, 15% - 20% -- but grade it generously enough and/or drop enough problem sets so that the students can see that (i) they need to spend significant time and effort on the homework and (ii) the homework that they themselves can do and turn in is earning them good enough grades.
It takes some skill to successfully use the web to answer your questions. The average young university student does not find the web the miraculous, Borges-ian answer book that those of us who have spent years studying our subject and honing our google-fu do.
If you hang out at a subject-oriented SE site like math.SE, you will be surprised how many students ask questions for which your tempted first answer is to include a link to a wikipedia article. But these questioners often clarify that they don't understand the wikipedia article / weren't looking at the right part of it / didn't understand why it answered their questions (in cases where it is immediately clear to the trained eye that it does). It's easy to forget how fragile your knowledge and understanding is when you first start out learning a discipline.
This has several implications for undergraduate teaching. (One of them, relatively little explored, is that we should probably be teaching our students how to search for information on the web. This is certainly an important skill...) One implication is that two questions which the instructor will regard as "isomorphic" (for the non math people: essentially the same, but perhaps superficially different) will not necessarily be regarded so by the students. For instance, when teaching (non-honors) freshman calculus class one can use webwork/webassign to give students various problems. Often these problems are generated from a much smaller class of template problems with some parameters randomized for each individual student. This is already enough difference to prevent students from easily doing each other's homework. But if you take things one level higher, then you'll see that most of what we ask students to do in freshman / sophomore level classes is to be able to solve a type of problem given a certain template. As a calculus instructor, I no longer have to think of "new" min-max or related rates problems: the internet has plenty of them. A student who combs the internet trying to get hints on which min-max problem is going to be on the test is quickly going to find out that if she can solve all five sample problems appearing on any one webpage or problem set then she can solve most of the problems that are likely to appear on the exam.
The best way to generate unique questions and coursework is to take a unique approach to the course.
When I teach courses at the advanced undergraduate level and beyond, I often type up my own lecture notes, which leads me to present at least some of the material in a different / new / nonstandard way. Having done that, it is easy to ask questions which are nonstandard. And any given undergraduate course (at least in mathematics, but I'd be surprised if other subjects were much different in this regard) can be taught in many different new / nonstandard ways.
With regard to what I said above, whenever I do something in my course or course notes which I think is "new", I then go the internet to see to what extent it is actually new. More than half of the time I can find something which I recognize as being an essentially equivalent idea or approach...but again, what seems "isomorphic" to me probably will not seem that way to a student first learning the material. A lot of times I find the past precedent in some article or note published up to fifty years ago. I am pretty sure the students are not reading such things at all: if they were motivated to try to do so in order to get a jump on assignments, that would be fantastic!