Keep the offer at 100K. If he's no better than the others, you can find another one. If you keep missing out on hiring because other companies pay more, then you need to raise your standard salary, because all your other engineers will jump ship within a year or two.
But what if it had already cost $10k in resources to recruit him and get him this far? What if it costs $10k to find the next person who will accept $100k? Then it's a wash, and all you've done is lost time.
It's only a wash if your only gauge is money. If you hire him, you've potentially sacrificed moral. Sure, it isn't immediately meassurable in dollars, but it is still an important price to pay.
Well these engineers weren't savvy enough to figure out that they could make 110K right now. Who's to say they will in two years? Or maybe you just give them a raise to 110K in two years and they'll be happy with that and never figure out they're worth 125K now.
The parent said "if you're having problems filling the position". If you're having problems filling the position because other companies are paying more that's going to come out sooner or later.