After testing the possibilities of adjusting LyX's behaviour using a "portable" approach as given in the other answer, in this case the solution was to simply add a fresh username composed of simple characters.
NOTE a number of users may arrive here who have either accented names or be composed UTF-8 characters or simply have a two-part name with a space. In such cases it is best to consider adding a new user to the system with a single plain ascii moniker.
e.g. c:\users\me\
IF you have already installed TeX and LyX you have likely encountered the issue of LyX complaining it cant find the TeX classes. So as a minimum uninstall LyX for that problem account.
Now using the new account BEFORE you do a fresh LyX install, first check that your TeX is correctly configured. On windows you need to confirm that the two following commands agree EXACTLY.
So to test all is well check that there is only one pdfLaTeX and that it is included in the path variable so from a command prompt run
> where /r \ pdfLaTeX
> where $path:pdflatex
both should agree it will of course vary for each user HOWEVER it should be one IDENTICAL entry for both commands
e.g. c:\ ..tex.. \ .. \some\bin\pdflatex.exe
Using the fresh account next check if installing LyX clears the problem. If you still encounter issues with the underlying TeX (MiKTeX or Tex Live) then you may need to consider re-installing one of those again. But again ensure you pass the where test before starting the new LyX.
When you make significant changes to the underlying programs especially MiKTeX updates, you may need to frequently use the "Reconfigure" option within LyX.
An associated problem can be the choice of PDF viewer since LyX uses background processes such as DDE or Pipes to control the pdf viewing. You may wish to check out alternatives.
Best Answer
You could do this from the menus with LyX. Use:
And get this pop-up window:
And input your symbol that way (like Word's ALT-I, if memory serves). You then can copy/paste if you want to keep using that symbol.
You can also use the "Evil Red Text" technique as noted by @AFeldman. If you use that approach, though, you will get the raw code in your LyX window, rather than the actual character (using đ = U+0111 = dec 273):
which produces:
I find it helpful to see the character! If you only use a few characters like this, you can create key bindings for them. Or if they appear in some national keyboard layout, you can also set these up as "languages" in LyX, and use the OS's keyboard driver for input.