Negotiations can be a tricky thing, and sometimes people have entirely unrealistic expectations that you simply can't fix. Typically this comes from people just starting off their career and go... programmers entry level make... and completely miss that location, market, etc are all HUGE factors in what "average" is.
I know someone like me who gets paid more
What someone else gets paid has little to no relevance to the negotiation at hand. Did this person apply to that company, get an offer, waiting to accept that offer based on your choice here... if not than how is their friend's success relevant?
The average pay is higher
This has "some" relevance, but only in two primary factors.
First, in your market, in your region, at their experience level what is average? If you're simply below that likely your employees will eventually figure it out and as spots open up in those companies you'll lose people to those companies.
Second, if other companies hiring for that role offer significantly more, retention will be very challenging as people figure that out. Sometimes you really have no choice but to accept retention issues if the role isn't worth as much to you, but if you low ball employee retention gets messy.
I can get a better offer elsewhere
Fine, go do it. No seriously if I'm offering you 50K and someone else is offering 75K for the same work you'd be stupid to take my offer. (we're assuming all things equal here)
I've actually said almost those exact words to people. "Oh, you could get paid more at Name of company? Then you should go do that. Unfortunately this role just isn't worth that much to us here."
What is reasonable and "fair"
This is ultimately what should be the driving result of negotiations. Both sides need to do their homework here.
You need to know well in advance what sort of return you'll expect having this person in your office. This gets extremely tricky to calculate in some industries but the short version is what revenue does this person generate, what expenses does this person mitigate, and ultimately at the end of the day how much of your profit would this person generate? (It get's WAY more complicated than that with nuances)
Once you know where all the money works you can safely say, this role will make me ~amount of money you can only justify paying the person less than this amount by a fair margin. (because as a company I hire people to make more money, otherwise my money can be put to better use else where)
If you figure this all out and what they are asking falls within the "profitable by a fair margin" range than what they are asking is realistic and fair. (of coarse many companies will pay as low as they can get away with) but we're talking maximums here not minimums.
My other employees only get paid...
This also is only partly relevant... it's relevant only if what's being asked is fair and reasonable when considering the return on investment of hiring this person is significantly more than you're paying your people. If that's the case than you are paying below what is fair and that's good for your company, but sooner or later others will offer more and your retention will suffer.
On the other hand paying this person more isn't a good idea since it'll upset the others and potentially cost you jobs as well. (same reason, just different triggering event)
How to tackle reality
Ultimately we have one of two possibilities here.
A. what this person is asking is technically fair and reasonable based on expected return on investment.
B. what this person is asking is not fair or is unreasonable based on the expected return on investment.
and ultimately which of these two scenarios it is winds up being irrelevant in your current negotiations... In both cases your best bet is to stick to the ranges common at your employer and accept there is a strong possibility you'll fail to retain this person. If you're lowballing them they are perfectly justified going else where, if you're not lowballing them they'll either accept it or they'll move on.
Long term negotiations
This is not a one time occurrence, there will be many others like this person in your future. You need to have a good idea what people are REALLY worth to you. You should make sure your people are paid / treated well enough to maintain a reasonable employee retention, without exceeding your cap. (where the sweet spot between good retention and the cap is varies greatly based on company, location, your employees, etc.) If you go too low you risk possible running loss. (one person leaves which leads to others leaving, chain reaction) if you go too high that's lost profits. (which is good for the economy but bad for the company)