lyxpreamble
I'm trying to add something to the preamble, but don't know how. I tried Insert -> Latex Code and just picked places on my article, but I guess they were never in the preamble? I guess I cannot just put code into my article.
I can view source under View -> Source. Can I edit source? It is not under Edit.
As egreg mentions in a comment, the reason you get the error is that neither the amsfonts nor amssymb package, where \mathbb is defined, is loaded. To force LyX to add one of these you can do one of two things:
Go to Document --> Settings --> LaTeX preamble and add there either
\usepackage{amssymb}
or
\usepackage{amsfonts}
Go to Document --> Settings --> Math options. Uncheck Use AMS math package automatically and check Use AMS math package, as in the screenshot below. LyX will now have added
\usepackage{amsmath}
\usepackage{amssymb}
to the document preamble.
The first of these, the automatic one, adds these packages only if you use some of their features through the LyX GUI. E.g. if you do Insert --> Math --> AMS Align, the amsmath package will be added, and if you e.g. add some text with Blackboard bold via the math font button on the toolbar -- 
-- then amssymb will be added.
There is a module that uses the sidecap package available from http://theorie.physik.uni-konstanz.de/jrioux/lyx/ (via http://wiki.lyx.org/Layouts/Modules). Download the sidecap.module file and follow the instructions on the that site for installing.
In LyX, go to Document --> Settings --> Modules, and add the Sidecap module which now should be available. To add a figure with a side caption, simply choose Insert --> Float --> Figure (side caption). Similarly for tables.
I'm afraid sidecap doesn't change the behaviour of normal figures, it only defines new environments where the caption will be placed on the side. To use sidecap I think you will have to add these environments "manually", in ERTs.
After adding \usepackage{sidecap}, you can do the following where you want to add a figure:
\begin{SCfigure}.Add a second ERT after the image, and type
\caption{<Your caption here>}
\end{SCfigure}
An example of how it may look inside LyX:
If you want to add a cross reference to such a figure you have two options I think:
\label{yourlabel} after the \caption{}. The problem with this option, is that the label doesn't show up in the LyX's GUI, so you'll have to write in in manually.\end{SCfigure} in a separate ERT, and in between this one and the one with \caption{}, add a label with the button on the toolbar or Insert --> Label.
For both cases, instead of writing the caption in an ERT, you could also add it using Insert --> Caption.
Best Answer
LyX hides LaTeX Preamble under the menu option Document > Settings... > LaTeX Preamble.