PCPaul wrote:
Because pneumatic or mechanical delay switches
with outdoor fitting kits are extortionately expensive,

Given any thought to how you will make *yours* weatherproof?
Priced an appropriate enclosure and fittings?
Ever built anything along these lines
(low-voltage control / mains on the output)?

Ideally a single button press should start a delay timer
that activates a light

A solid-state relay seems like the easy way for a 1-time project.
Logic-level inputs and whatever-you-specify on the output.
A parametric search at your favorite parts vendor's site
should turn up a likely candidate.

for a few minutes
(to light the way from the garage to the house).

This is the way most folks get a delay of over a minute:
http://www.google.com/search?q=CD406...ator+datasheet