What is the name of the default font in math mode?

The font is a math font that is not available as OpenType. No, wait: it is! There's the new "Latin Modern Math" that's in the new TeX Live 2011 (to be released in a few days). The font is here. I haven't seen it yet in CTAN, though. The default font is Computer Modern, but the differences with Latin Modern are negligible for your needs.

Here a way to find this out, even when you change your font later:

For a PDF LaTeX document with only $i$ as content (and empty pagestyle to avoid the page number) I got using pdffonts the font name ZTAVXI+CMMI10 (the CMMI10 is the important part AFAIK and stands I presume for "Computer Modern Math Italic 10pt"). My full TeXLive2010 installation included only the TeX fonts, but apparently LyX delivers it as TTF font, at least I found it under /usr/share/fonts/truetype/ttf-lyx/cmmi10.ttf with my Linux installation. You might need it however as OpenType font. AFAIK it might be better to not use the original default font but a newer replacement like lmodern (load that as package) or the font egreg mentioned in his answer.

Full Computer Modern Math collection is available here as .otf files for use in any non-LaTeX programs suporting OpenType fonts. Same web page have also Computer Modern available.