Keep one's hair on indeed means
To stay calm or, to be patient.
History
As far as I know, it's a colloquial British English idiomatic expression, urging the other party not to lose their cool. However, it seems to have spread across the globe and is widely used across the US, Australia and other English speaking countries.
Acc. to Google Ngrams, the phrase first appears in 1868-69. To keep your shirt on has the same meaning and tone but doesn't seem to appear before 1870-71, according to Google Ngrams and in 1904 according to Etymonline.
NOTE: The expression "keep your hat on" predates them both to the year 1804.
Extended Explanation
(Disclaimer: This is some sheer guesswork, putting two and two together.)
The idiom seems to be constructed from that fact that one might lose their hair due to stress (check this), or even might pull it out in exasperation, anger or frustration.
The Ngram results definitely indicate that keep your hair on is currently more popular than keep your shirt on". And for what its worth, these expressions have nothing to do with to let your hair down.
Best Answer
I have to agree with Fumble' s comment. This is likely a conflation of information your wife is performing.
Typically it's either -
Keep your pants on: means to wait patiently
or
Hold your horses: slow down, hold on, wait, be patient
or alternatively
Cool your jets: slow down, hold on