Cute, a 3-terminal sync reg!
So inductor ratings, low enough resistance that it's not annoying of course (within thermal ratings certainly, and any efficiency gain you want on top of that), and saturation current >= peak current used. So, max DC (again, of the regulator!) plus peak ripple (or half of peak-to-peak or delta I).
In that size, multilayer ferrite is probably fine, otherwise most anything wound ferrite or composite powder should be fine. Small chip inductors come in all sorts of styles, take your pick. Raw-ass ferrite beads wouldn't be suitable (not that you'd find one with so much inductance, nor sold as such; multilayer FBs are made exactly the same way though, so, just that you won't know by eye in that case, don't let them get mixed
), and powdered iron toroids can be an issue as far as losses (not that they'll be a problem for immediate purposes either -- you'd be pondering those for 100s W capacity converters). Shielded is usually better than unshielded; it kinda doesn't matter for tiny stuff like this, but is nice for low noise, and power stuff where not only does it reduce emissions but an open-core inductor with a lot of voltage on it might start cooking (induction heating) neighboring components.
Unfortunately the datasheet doesn't really tell you what or what things need to be what they are. No internal compensation is listed. PFM could be as simple as hysteretic control again, or it could be an independent oscillator skewed by operating conditions. Also not obvious if it's synchronous rectified, or just free running (always driving LX high/low); efficiency seems reasonable to lowish currents so maybe it's the former. It could also be a burst mode control, with consequent higher ripple. It does say very low current at no load, suggesting some kind of burst mode. But no way to really tell from here. Such is the downside of shopping in China...
That they suggest tantalum, implies they want ESR. Electrolytic can be used as well, just shop around for something comparable (0.1 to 1 ohm ESR?). Ceramic can also be used with external ESR. If it works fine with ceramic alone (do a step transient test -- again, a scope is extremely useful here
), well there you go.
Also, you probably want one a little step up from that, with only the 6V rating? Since you were looking for 5.9 or so. I suppose it doesn't matter as it's fixed 5.0V anyway, but more might be nice for the overhead you wanted.
Tim