What kind of riding is he doing? Commuting, trail riding, or road riding? 5’4" isn’t ridiculously short, and he should be able to find a bike to fit him. I’d definitely choose the style of bike most appropriate for my needs and then go from there.
For a mountain bike being used on trails, one really wants to be able to stand over the top tube and lift both wheels a couple of inches above ground. I’m all of 5’6" and have a sloping top tube 13" frame that I really like.
Of course it would be the best idea to go to a bike shop to confirm the fit, but the answer to your question is yes, a 16" frame should be right in the ballpark for someone about 5ft, 4in tall. Like The Conqueror Worm says, you should be able to straddle the top tube with a couple inches or so to spare.
I’d say he should be looking for a city commuting bike or a mellow cross country mountain bike. I happen to be a fan of Specialized so let’s look at their offerings. Most major manufacturers will offer similar bikes.
For city bikes, the low end is a Crossroads Sport at $470. It’s got 700c tires and the smallest frame has a standover height (I’m unsure of how this is calculated) of 724mm (28.5") and a seat tube length (top of top tube to center of cranks) of 370mm (14.5").
For mountain bikes, the cheapest is the Hardrock at $390. The seat tube length is 330mm which is 13" and a standover height of 27.4".
Both of these bikes are pretty small and I suspect he’d be happy with either.
Personally, I’d go with the Hardrock and stick on some narrow slicks or dual purpose tires (the local bike shop will probably be willing to swap them for free). Then I’d get a set of knobby mountain bike tires in case I ever wanted to go offroad.
Giant Bicycles is a super huge Taiwanese manufacturer that probably builds the aforementioned low end Specialized bikes. Have a look see and see if anything appeals to him.
It seems that most manufacturers prefer to put a suspension fork on entry level mountain bikes, rather than rigid. Probably for marketing reasons - makes it look more like an off-road machine. Whilst the low-end forks tend to be rubbish compared to something good, it’s not a big deal if your friend is just getting into the sport. It won’t really disrupt the riding he has in mind.
Yes. There is also a less common 650c size intened to work better on pavement than the 26" size, yet be a bit smaller than the 700c bikes. Bikes, tires, tubes etc. will be a bit harder to find in this size, but may be worth the hassle for some.
Mountain bikes make OK “city bikes” for short trips, especially if you put smooth tires on them…knobbies have very high rolling resistance. BUT look into “hybrid” bikes if most riding will be on nice surfaces. Efficiency of a road bike (mostly) combined with ergonomics of a mountain bike.