i used to have the sony dhc h1 but went i got it, it had been out for a month only so it was full of bug , i mean i had to return it to the shop 3 times... now i know it is working properly, sony fixed all these small errors.
i have the Sony DSC-P200, i just love that camera, i mean you can do so many things with it it's incredible, i would actualy for this one instead of the dhc h1
or if you really wanna go for quality , i would gor for that one:Canon PowerShot S2 IS... it is the first camera that can even focus on the dirt of the lens in macro mode... i would deffinatly got that one, but i didn't have enought money.... as tony said go check them on http://www.dpreview.com/
, i used that website to get mi camer (to chose it then i bought it in a shop)
on thing, i would not get a Fujifilm one because the menus are really complicated to use.... i love the sony menus, they are nice and it's great if you want to change a setting very quickly... personaly as i said, i have the sony DSC p200 and you can set so many differant things with it... it's crazy. have a look for example at these pics i took with it http://www.richardadenot.com/pic.htm
i would either go for the Canon PowerShot S2 IS or the sony DSC p200 (that camera really lloks like a tourist camera but it isn't, its just like a angel stock in a trash can....
hope all that info was usefull