Is there a way to create a new symbol and add it to those made available when clicking on the omega sign?
I am not aware that LO can change fonts. For this you need a separate font editor.
Ebot, yours is a valid answer. Repost as one. Yes, LibreOffice can only use symbols available in fonts that it can find in the system. Thus, to have a new symbol available you need to edit an existing font or create a new one using a dedicated program such as FontForge. In case of certain fonts, especially free ones developed by communities (DejaVu fonts are an example), you can file requests for new glyphs but without any guarantee.