Does anybody know a font on Google Fonts that is very similar to Helvetica Neue?
Arimo (see samples above and below) is identical to Liberation Sans (suggested by Yisela) and available from Google Fonts.
The following font stack includes Arimo and suppresses the substitution of Arial for Helvetica on Windows machines and WebKit browsers, but otherwise puts the most Helvetica-Neue-like fonts first:
"Helvetica Neue", HelveticaNeue, "TeX Gyre Heros", TeXGyreHeros, FreeSans, "Nimbus Sans L", "Liberation Sans", Arimo, Helvetica, Arial, sans-serif;
For further explanation, including the advantage of inserting "Microsoft Sans Serif" ahead of "Liberation Sans" for large non-italic headings, see "A multiplatform Helvetica-like font stack that suppresses Arial".
The choice between Arimo (above) and Roboto (suggested by DavidO) is a matter of taste. Roboto loads slightly faster, but Arimo doesn't need to load at all if Liberation Sans is installed. As both Arimo and Roboto are Google fonts, there's no point in using one as a fallback for the other.
That's the thing about Helvetica, there isn't really anything quite like it. We are talking about one of the (if not the) most beautiful font in existence.
You could settle for Open Sans or Source Sans Pro but it won't be the same. But you probably already know that as there are only 121 fonts to choose from.
It's not in Google fonts (I think), but the .ttf can easily be converted for @font-face. I find Liberation Sans really beautiful. Plus it renders REALLY well in most browsers, and it's free for personal and commercial use:
I just tried the "Roboto" font and it seems OK: