How to insert slashed o "ø" into an author's name in my bibliography

When typing the name in the text, N\o rregard will not leave any space in the output, as spaces after control sequences (with name consisting of letters) are ignored.

However, in .bib file the question is slightly different, as you want to use the name also for collation. The BibTeX manual recommends

author = {N{\o}rregard, X.}

because in this way the entire combination {\o} would be regarded simply as an "o". The typeset text will not have font dependent kerning between "N" and "ø" and between "ø" and "r". If you're using biblatex, then Nørregard is fine (as long as you use an input encoding where ø is present).