Just saw that getting from Gatwick to Waterloo this summer to catch the Eurostar to Paris might be complicated by underground construction. Does anyone know more or have suggestions?

No there are no major problems getting from Waterloo to Gatwick as they are several different routes available; www.thetrainline.com for schedules. You'll probably have to change once by train.

What construction?

If you're talking about the Waterloo and City Line, it's irrelevant to your planned journey.

Personally, I'd get the Thameslink to London Bridge, then a cross-platform to Waterlo East. Bit of walking, but on the level all the way. Or Southern to Clapham Junction, then any train that's going to Waterloo.

I wouldn't bother with a route involving any tube line.

Interesting, that change at London Bridge sounds like a better option for us getting to Covent Garden (taking a taxi from Waterloo) than a train to Victoria then a taxi from there. Right? How much is a ticket from Gatwick to Waterloo?

(a) Yes, Waterloo is closer to Covent Garden than Victoria
(b) Pricing is usually to "London terminals", i.e., it makes no difference if you get off at London Bridge or Waterloo: from memory, Gatwick to Waterloo via London Bridge would be a few pounds cheaper than the Gatwick Express to Victoria but probably about the same as the local service to Victoria.

If I were going to Covent Garden from LGW, I'd go past Waterlo East on the train from London Bridge, and on to Charing Cross (same price).

Then walk. Or if luggage was a real pain, cab (though expect some GBH of the earhole). Apart from anything else, getting a cab from Ch X is a lot easier than at Waterloo East.