Depending on the situation you have a few options:
1) The Region Group tool will allow you to isolate individual pixels, which can then be filtered based on count.
2) The Con tool can be used to remove (set null) or change the value of pixels if you can identify them by their values.
3) A variety of filters can be used to remove isolated pixels (e.g. majority filter). You can combine this with a mask if you don't want to filter the entire raster.
4) If you're looking to just display the raster you can just clean it up in something like Photoshop.
In terms of manually performing these types of edits, this related question (including solution) is probably your best best given that you can't just edit individual pixels with something like a simple toolbar.