I dealt with something much like this recently, where working for an insurance company led to a background check that needed specific days of start and end work and proof of income from jobs I had over 7 years ago - it's a real fun experience. Or not.
In my case it was simply because they couldn't talk with the owner of the business to verify my income because the company was sold. That didn't matter - to approve me they needed verification none the less.
Note that this isn't an inquisition (even though it can feel like it), and this isn't all that abnormal. The main problem you can have is if someone made a "material misrepresentation". Otherwise it's just going to require the appropriate verification be made, so the background check can be completed and approved. Be in contact with them and cooperative and explain any discrepancy, and you'll likely end up just fine!
Note also that job offers are often contingent on an approved background, so don't fret that the job is on hold - that's normal.
The question is then, what do you need to do to provide proof? Amazingly as I found out, there's lots of things you can do if a simple "letter of explanation" turns out to not be sufficient!
One thing, and often the first line of proof, is information from the IRS. If you somehow still have the original W2s (W2 information being provided was mentioned), hat tip to you and that's great. But if you don't you can still get an IRS transcript that will at least show your income going back 10 years (or possibly more), and this can be used to verify the past income.
If that isn't suitable, one can also get a Social Security Administration Earning's Information report that can give income and addresses of employers going back...well, all the way to when you first worked a job that reported to the SSA. This takes time and a small fee, but it again can be used to prove income even if a company is unavailable, is wrong, or if they just make stuff up.
Like the good book says: Don't Panic.
It's perfectly natural for it to be a scary and worrying experience, but many of us have been in the same position before and it's OK. Even if a reasonable mistake was made in entering old job info, you'll be asked to explain and if it isn't deemed "material" then it won't effect the job - just hold up the background check and rile your nerves.
Stay in contact with the employer and the background check firm and ensure you are all on the same page - there was a hold-up in the background check but you are working to resolve it and provide any documentation requested, etc. Pretend to stay cool, even if you don't feel that way. It'll be resolved in a matter of days or a few weeks, generally - which is terribly unpleasant, but then it'll be over and life will go on.