I would personally suggest finding someone in your local area that can string pearls well. Knotting pearls well takes practice, and if you only have a few strands that need upkeep, you would be better off with a professional stringer. Treat the strand like the cleaning of your fur coat (faux fur coat - sorry, the Angelino coming out in me), about once a year. Unless you have a keen interest in beading, the $10-$30 you will spend on the reknotting will outweigh the cumbersome task of learning to string.
Silk is not as important as many think. In fact, the large majority of stringers "double knotting on silk thread" are either using nylon or a silk nylon blend. I do not really believe one is better than the other - one just sounds better. Almost anything coming out of Asia (both Japan and China) is not strung on pure silk.
I totally agree with Jeremy's post.
Dear Pearl-man, I agree with you to a point, Quality is paramount, but Silk of itself doesn't define or guarantee it - and some of the "Synthetic threads" of today are a lot different to yester-year.
Like You, I have been stringing professionally for many decades - I wont give the year as is a Ladies prerogative
Also like you I was taught on the Japanese Tire Brand silk thread, and then through circumstance was introduced to a blended synthetic thread from Kaygee which tested extremely well - eventually becoming a "Go-To" staple. I now also use Pattye's Secret Beaders Thread in colours for a good 7 years (Bless Her)
Both These threads drape Beautifully and are in no way detrimental to the pearls. Those who know me in the trade here in Australia are quite aware how particular I am regarding thread and Gimp (Gimp is another story of its own). I would not hesitate to thread a premium strand on these threads - and indeed have done on strands valued around up to ~150K
As a footnote to the silk - After years of using "Alternates" I went back to Silk for a strand of South Seas on the customers special request and had my first return in a very long time with a strand that stretched.
There are Good Bad and Middling synthetics, just as there are Good Bad and Indifferent Silks (e.g. Filament Silk Versus Spun) Natural products suffer(?) their own variations in quality by the very nature of being natural.
Another Variable, and a very important one, is the skill and technique of the threader. Should they have a grip like a monkey and over tension the thread and knots, or head the other way with insufficient or uneven tension, the result will be a bad drape and lifespan of the job, whatever the material.
Good Silk has some (many) good qualities - but also has some inherent deficiencies.
A Good Synthetic can be indistinguishable - in regards to the qualities relevant for pearl threading - and at same time avoid the deficiencies of the natural product.
In summary - Quality Thread AND Technique are Both needed for a "good" rethreading job.