Will adding yogurt whey to homemade mayonnaise make it keep longer?
This is generally referred to as "lacto-fermented" mayonnaise.  The whey is assumed to have active bacteria, and most recipes insist on a room temperature rest for at least several hours.  During this time, I suppose the assumption is that the bacteria from the whey will ferment and produce sufficient acidity to act as a preservative (as in sauerkraut or something).

The idea is not completely without merit.  As discussed in my answer to this question, there is a well-documented and validated food science procedure for killing off harmful bacteria in homemade mayonnaise, which involves sufficient acid content and a rest at room temperature before refrigeration.  Mayonnaise that follows this advice should be roughly as shelf-stable as commercial mayonnaise, and its acidity will make it a poor growth medium for most harmful bacteria.  It will generally deteriorate in quality not due to spoilage bacteria, but due to oils going rancid and other such processes that will degrade texture and flavor over a period of weeks.

On the other hand, I have serious doubts that a few hours at room temperature with a tablespoon of whey is going to produce a consistent level of acidity to ensure safety and longer preservation.  You simply don't know the bacteria count and activity in the whey, how well it may grow (if at all) in the mayonnaise, and how much it will lower the pH.  (In a quick search, I couldn't find comparable food science research on using lactofermentation to achieve safe homemade mayo, but it may be out there.  It sounds implausible to me, not least because of the lack of sufficient food in mayo for the lactic fermentation to produce a lot of acidity.)

If you want to make safe homemade mayo, you need sufficient acidity.  Usually that comes from lemon juice or vinegar.  It's that simple.  And with adequate acidity, it will likely have a reasonable shelf life of more than just a few days.  There is further advice and thoughts on that issue in another question.  But I would not recommend this whey-based technique to ensure food safety.

EDIT: Despite the huge number of recipes online for lacto-fermented mayonnaise, I've been unable to find any reputable food scientist vouching for even the possibility of significant fermentation occurring in mayonnaise.  Nobody reputable even discusses it (probably since, as I said above, it doesn't make a lot of sense).  Anyhow, I did manage to find this, in a list of fermentation "myths" maintained by a prominent fermentation Facebook group:


  MYTH: Mayonnaise can be fermented by adding a bit of whey or
  sauerkraut juice and letting it sit on the counter for [x] hours.
  
  FACT: Oil cannot be fermented, and mayonnaise is primarily oil with a
  small amount of egg yolk emulsified into it. All bacteria have an
  absolute moisture threshold necessary for their survival and function,
  which mayonnaise doesn't provide. . . . The amount of moisture that is available for use is called
  the Water Activity level, or aW. Pure water has an aW of 1.0. Lactic
  acid bacteria require a minimum aW of 0.94, but not much activity will
  occur below 0.95 and a level closer to 1.0 is needed for vigorous
  growth. Mayonnaise has an aW of 0.93 or lower. . . .


I wouldn't say this qualifies as a reputable source for food science (and I haven't checked the facts), but the reasoning is plausible, as bacteria simply don't grow well in mayonnaise.  If anyone can find any better food science on topic, I'd be interested.