Can I replace a single watt (100 max) socket with a 3-way socket on a table lamp?
Yes, you can.  The sockets cost about $4, they are pretty standardized, and it requires "competent handyman" level skills.  

However, the "100 watt maximum" is to be taken seriously.  That is a thermal limit for that fixture, and the fixture could start a fire if a larger bulb is used.  For purposes of this rating, this refers to the genuine, absolute, true wattage of the bulb (e.g. 26 actual watts for a "100 watt equivalent" CFL, or 16 actual watts for a "100 watt equivalent" LED) — not the actual light output (typically 1400-1600 lumens for a "100 watt equivalent" bulb.) 

So for instance if you found a hypothetical 5000 lumen, 48 actual-watt, "350 watt equivalent" LED bulb that you just had to use in this thing, then yes, that is fine.    However, a 50-100-150 actual-watt incandescent is Right Out.  

I splurged on a $17 GE LED bulb that is 50/100/150 equivalent actually about 25W and I am in love with it.  Interestingly, its sequence is Off-Lo-Hi-Medium, and they do that on purpose so if you stick it in a 1-way socket, you get the full "150W" equivalent.