Huh, what is a "new" engine? A really all new engine for large airliners has hardly been designed for 30 or 35 years.
I would consider the following engines "new" designs: P&W JT9D, GE
TF-39, RR RB
F101, P&W F100. And to that we can add half of a new engine, the SNECMA M56.
They can be considered the grandfathers of present day PW
CF6, GE90, GP7200, RR
Trent, CFM56 and (maybe slightly less) the IAE V2500.
If we have to pick one really new engine during the last 30+ years, then the ill fated PW6000 probably comes closest. Its development went years and billions over budget.
When the first A380 took to the air a couple of years ago on it's four RR
Trent 900 engines, then those engines had been under development for five years, but were still considered preliminary test engines. A substantial amount of hardware, things which should make those engines really operational in everyday use and give the planned redundancies in failure modes, were not available yet at that time. The Trent 900 it a little more than a refined Trent 800 since it's the first triple spool engine in the world with contrarotating core.
You mention GEnx. It's definitely a refinement and development of the GE90.
Like with cars, different and improved models are produced every now and then. Sometimes it gets a new name, sometimes only a different numeric or alphanumeric code as a suffix to the name. For instance one of the more massive changes to an existing engine was some 20 years ago when the CFM56-5C2 was produced on the basis of the CFM56-5A3. Other engines have got a new name based on much less change.
But in any case, when another engine variant is produced, based on substantial hardware changes and improvements (like for instance GEnx or RR
Trent 1000/1700), and not just FADEC software updates based on a previous model, then a billion dollars are hardly peanuts, but much less will it cover a total bill. Development and certification takes several years.
I doubt that there are airliners out there on which the engines are fully 50% of the procurement costs. Most will have the engines in the 25 to 33% range. But in any case those hair dryers are not cheap. Just one GE90-115B will probably retail around $20 million.