Good question and I just ordered a new under table router to replace my Bosch 1611, 15 amp plunge router. My 1611 is similar to the current 1619, a big router. My 1611 is 16 years old and has no modern features other that it is electric. Anyway, what you want is a router that you can fine tune the height adjustment with some sort of adjustment knob after making your quick coarse height adjustment. Note my mentioning "after" making your Quick coarse adjustment. You do not want one of those after market threaded tubes with a knob on it. Those take forever to run up and down from one end of adjustment to the other end. Something to look at for example is the adjustment setup on the newest Milwaukee routers the Bosch 1613 or 1619 or the Triton router. All of these routers and I am sure a slew of others will work nicely under the table. Some are fixed base, some are plunge base. Your main concern is to be able to make coarse height adjustments quickly and then be able to fine tune that adjustment. I also have a Bosch 1617evs router that would work great in my router table but it is kinda small for some of the bits that I spin on my router table. If you are turning common horizontal raised panel bits I strongly suggest one of the 15 amp "aka" 3+ hp routers with VARIABLE speed. You need to slow those big bits down to the
10,000 rpm range.
So today I went down to buy my 6th router and had narrowed the choices down to the fixed base 5625-20 Milwaukee or the plunge base Triton. I had considered the Bosch 1619 evs but because of the location of the fine tune adjustment knob it would be behind the router when turned up side down or on the left side of the router. Neither of those locations appealed to me. The Milwaukee is a beast and well made. Simple. Variable speed, fast coarse adjustment, good fine tune height adjustment knob and the ability to make fine tune height adjustments "above" the table. It is also a 2 wrench router which I GREATLY prefer over the single wrench models. BUT, I found that when the Milwaukee is upside down and you push the coarse adjustment knob to make an adjustment, you better have a both hands on the router to keep it from falling out of the fixed base. While the above table height adjustment is a cool idea I can see the hole in the table top filling with saw dust and clogging the mechanism even though there is a plastic cover that opens when you push an adjustment tool through it.
So now I turn to the plunge base Triton, Seems to be designed from the ground up to be hung under the table or used free hand. While I have read many mixed reviews about 3 or 4 specific problems, these apparently have all been addressed and the new production Tritons have been fitted with the fixes. I ordered the Triton from the Woodcraft store. They were sold out. The big pluses I see here is that bits can be changed from above the table top. While this is a 1 wrench router like my current Bosch 1611 the router will remain in the base during a bit change. While 1 wrench routers sound simpler to use, you still have to use your other hand to hold the router and or the router shaft lock. So what have you gained by only using 1 wrench? With 2 wrench routers, you can loosen or tighten the collet with 1 hand. 2 hands are still needed to make coarse adjustments to the collet. The Triton having a self locking shaft and not requiring removal of the router from the table may be somewhere in between the 2 wrench and 1 wrench style as far as ease is concerned. I suspect one hand will be on the wrench and the other holding the router cabinet to keep it from moving when loosening or tightening the collet. The Triton will not fall out of its base when making coarse adjustments under the table.
Other nice touches that the Triton has over other plunge based routers are, the adjustable depth turret is second to none and simple to use requiring no screw driver or wrench for multiple height adjustments. This may or may not be useful in a router table. The Triton comes with an edge guide for free hand use.
Routers are generally plunge base or "non-plunge" fixed base.