Your choice for mid/late January is all snow-related. Snowfall over the Alps generally decreases from N/NW to S/SE, and the same is true for the expected coverage and snow bases during the time period.
	
		
	
	
		
		
			Which is the single best place from that list for a full week of skiing for my son and I? He's expert, I'm advanced. My budget would be moderate. My wife the non-skier might benefit if we stayed at an interesting base village, but that's not a make or break factor. Her time would come the second week.
		
		
	 
Given the mid/late January timeframe (and Ikon restrictions), I would most likely choose Chamonix.  Followed by Zermatt and maybe then St. Moritz.  Aosta next (why? Cervinia is essentially Zermatt; other areas receive less snow than Chamonix/one could ski Courmayeur from Cham). Last/likely a pass Dolomites and Kitzbuhel.
Why Chamonix?  High, reliable snowfall area (generally, the town will have snow cover in January, but not always).  Lodging at any price point. One town mountain - Brevent/Flegere, but otherwise by bus/car.  Easy to join a cheaper 'guide' group for off-piste or Aguille. Lots of nearby resorts that get good snow and others that are weather-proof (trees for visibility).  Again, the highlights are:
- Grands Montets (it is horrible that it has taken almost a decade to replace the summit tram - next year 26/27 S3! - but it's the equivalent of Jackson missing a tram for a decade.  Again, the current lifts can provide great skiing!
 
- Brevent/Flegere - solid intermediate to expert terrain, great views, face south?
 
- Le Tour/Tete de Balme - uncrowded open freeride skiing
 
- Aguille du Midi - Vallee Blanche and Envers variants are all global ski world classics! Issue:  glacial crevasses just filling in by January
 
- Courmayeur - easy Mont Blanc tunnel.  Classic town, and Dolce Vita scenic mountain with surprisingly great off-piste!
 
- Verbier - easy day-trip.  My favorite in all the Alps for pure, easily lift-served, steep skiing!  (rank it slightly ahead of St. Anton/Arlberg, Val d'Isere, Andermatt/Engelberg - other favorites)
 
- Megeve - a massive area with great tree-lined runs.  Better than Les Houches. 
 
- Flaine? La Thuile?  Huge areas with great snow potential.   
 
Zermatt and St. Moritz are strong contenders.  Both get high-quality snow, are super scenic, have great terrain, but sit deep in the Alps, almost on the backside of the Alpine Divide, and get less snow than other West/Northwest Alps resorts.  They are very similar to the USA's Aspen, Telluride, Crested Butte, and Sun Valley - snow can take a while to accumulate, and expert terrain often becomes available reliably after February 1st (or 15th?) most years.  Some years sooner.  For example, last year St. Moritz was <50% until late January and a 3-5 ft storm.  
If snow does materialize, I prefer Zermatt's experience and expert terrain to the piecemeal approach of the current Chamonix.  And no car is necessary for Zermatt.  It will just sit in the Tasche parking garage for $20/day.  Improves Zermatt's affordability index (3-3.5 hrs from Geneva/Zurich on a beautiful train ride).
If they have good early snow years, they might be better choices due to fewer crowds, less hassle getting to the lifts, an overall better experience, and more reliable snow.
	
		
	
	
		
		
			Discussion:
- I've always wanted to ski the scenic Dolomites, but I'm not sure I want to tangle with the place two weeks before it hosts a Winter Olympics?
		
		
	 
As Tony suggested, the Olympics are spread throughout the Dolomites, with events in Livigno, Bormio, and Cortina.  Like the Salt Lake Games, spectators generally show up, but in overwhelming numbers. And skiers were a no-show in Utah for the entire winter of 2002.  Will not be an issue.
	
		
	
	
		
		
			- Zermatt would be nice. I'm not sure it fits a moderate budget? We've visited in the summer 40 years ago and were blown away by the Matterhorn scenery.
		
		
	 
Zermatt is quite large, and you can find reasonably priced accommodations around the train station (north side) or on the south side.  Avoid the city center—same advice as St. Moritz (Dorf=expensive, Bad and Celerine = more reasonable).  Surprisingly, Ischgl is quite upmarket and priced more similarly to these resorts.
Again, the issue is not accommodations.  You can find lodging at a similar price to mid-market Colorado.  Keystone, Steamboat, Park City, and Winter Park are not cheap anymore!  Kinda picey.
Food is an issue - anywhere in Switzerland.  Breakfast is most often included.  However, lunches and dinners can add up!  Lunches are more often sit-down meals versus takeaways, and dinners are generally more expensive (with fewer casual pizza and brew pub places).  Combined with an unfavorable exchange rate compared to recent years (1 CHF = $1.25 USD, down from almost parity), the cost of food adds up.  
My Swiss workaround is (lunch is often Coop groceries, breakfast remnants, or Goulash Soup), and dinner is half-board.  As you can see in many reports, when I am not with friends, I mostly ski through lunch, and I will bring protein bars from the USA.  But will splurge or stop for a refuge or restaurant with a good view/location, or food reputation.
	
		
	
	
		
		
			- St. Moritz or Ischgl both sound like they could be good for a week-long visit, but I know little of the scope and quality of skiing at them?
		
		
	 
Both are excellent.  Lots of reports online.  St. Moritz has more choice since it's three unique areas (Corviglia (sunny, south-facing/cruisy town mountain), Corvatsch (big, burly, short bus ride), and Diavolezza/Lagalb (open, freeride, big vertical trams).
	
		
	
	
		
		
			-I get the impression that the skiing around Chamonix is somewhat disjointed. Is there excellent recreational skiing beyond the renowned route up the Aiguille du Midi and the Valle Blanche tour, which I'm not sure I want to do?
		
		
	 
Chamonix = St. Moritz = Davos = Aosta.  All are disjointed/unconnected with one town hill, and everywhere else is a drive or bus. Chamonix's buses can be jam-packed, and having a car is a nice option (Verbier, Megeve, Les Contamines, etc.)