I'm being a ASP.NET developer over years but recently I started to work with some PrimeFaces solutions with JSF 2.0. I haven't nothing to complain about it.
It is worth emphasizing a few points that did me likes the PrimeFaces: many components, easy to use, fully based on jQuery and jQuery UI (which are very important to me) and very focused on web standards (even more important to me).
But there are some cases that I would really want my server-side with ASP.NET. I don't want to start a discussion here about advantages between JSP and ASP.NET, both has many motives to be used.
Well, the standard ASP.NET components aren't jQuery based, aren't themable with jQuery UI Themes, aren't fully based on web standards (a big problem) and the variability is much more simple (there aren't Carousel, Accordion, Captcha, Slider, Spinner, etc, etc, etc).
EDIT
Continuing the saga, the @Raynos answers inspired me to study lighter possibilities, like the suggested manos. But I really would like something a little more automated, at least in the HTML generation for common controls. Someone known working CRUD samples for manos or additional modules supporting this task?
Recalling that the main question is still open anyway: Do you know any UI Framework for ASP.NET based on the same ideas of PrimeFaces, prefered open-source like it is?
I suppose we cannot answer your question with something simple. As showed by other answers, unfortunately we don't have in ASP.NET any UI framework fully web standards based.
You surelly have that choice about not using ASP.NET but another .NET Framework based engine. I really think that ASP.NET MVC would be the more closest to JSF and so to PrimeFaces. But it haven't a large set of builtin controls based on jQuery as you liked on PrimeFaces. Also, if you only used up to now the vanilla ASP.NET, the MVC concept can be very confusing to start through a professional project.
It may be safer to continue with your technology knowledge while learning something totally new. Even so I don't indicate manos to your project, although it shows to be very light and stable. Furthermore, with it you would need to "recreate the wheel" for many tasks that ASP.NET knows to do well within the standards and such.
Based on these arguments, it seems that in your case you (or your team) is so familiar with ASP.NET that your immediate need is to use good tools to improve the shortcomings of ASP.NET on HTML5, web standards and themes. Then you don't need to change everything you knows right now, just improve what you already know (and of course we recommend that you consider these other approaches gradually).
So let a few helpful hints for you:
class names and DOM hierarchy. If your PrimeFaces Complex DataTable gerenates an incredible HTML which is so beautifull with your CSS, just use the same generated HTML inside a UserControl so you can reuse it around all yours pages without needing rewriting all the HTML.Then, finally, there are many tools and approaches to support you enjoy what you've done with PrimeFaces and also enjoy your existing knowledge in ASP.NET. Unfortunately we don't have done something like this to ASP.NET, but maybe what you'll do will be so good that you can publish opensource for everyone?