I've heard this a few times, and I would presume that it comes from Wellingtons, with the meaning of put some boot to it.
Is there an origin for this phrase?
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You are all missing the point that Wellies were, and still are, worn on working farms and would indeed be used when driving a tractor. It's credible then that 'give it welly' would be shouted at someone driving a tractor, hauling a load or driving out of the mud and it necessary then to push hard on the accelerator in order to get out of the mud or shift the load. The phrase, in hiberno-english would have been originally used in a farming context or a building site for the same reason.
OED has the definition...
wellie/welly: slang. A kick, acceleration. Also fig.
...under the entry for Abbrev. of Wellington [boot], but I must admit when I use the expression "Give it some wellie!" I've always assumed I'm referencing a derivative of...
Verb: To beat, thrash.
Noun: A stroke with a lash or pliant stick; also, a heavy blow with the fist.
So unless anyone can produce evidence of "give it some boot" being an earlier/alternative form, I'm sticking with my current thoughts on the matter.
EDIT: On the basis of no evidence whatsoever, I'm now more than willing to give credence to the possibility that the origin relates to Welly Wanging, which could have either started or been gaining nationwide awareness around the late 60s / early 70s (the same time as the expression under consideration here). The art of successful welly-wanging is to impart as much acceleration to the boot as possible before flinging it.
There are also probably phonosemantic factors involved in the retention / spread of currency. Ordinarily, the wang in welly-wanging means "throw", but here, for example it's a welt, whack, wallop, wham, whop. Lots of w- words imply the application of force.
Give it some wellie appear in several searches to mean give it some gas or, colloquially in AE, put your foot into it (put your foot down on the gas pedal), step on it (step on the gas pedal).
Wiktionary UK sums it up similarly:
(UK) To increase fuel or power to an engine, as to a car by depressing the gas pedal.
(UK) To apply great physical effort to (something).
Here is a humorous blog on the subject of the origin of give it some welly, which points to the site called World Wide Words (you have to search for yourself it you use the link in the link given by Simms, but I've linked directly to the page). World Wide Words suggests the phrase came about in the 1970's, either in the motor racing world, meaning put your foot down on the accelerator pedal, or in football, meaning put some power into your kick.