How much mass would have to be added to the Sun to significantly alter
Asking how much would be significant is inexact. The Sun is classified as a G2V main sequence star. Though the chart lists one solar mass as G4V, so there's some variation in there. The classifications seem to relate to temperature. To go 1 step up (and using the sun or any G star), the increase of one ranking (decrease in 1 number after G) alternates between a 2% and 5% increase in mass. I'm going to round that to 3%-4%, so for our sun, that's about 30-40 Jupiters (31.4-41.9 if you care about significant digits), but I'm going with 30-40 Jupiter masses added for each step up the classification ladder, ignoring the effects of impact, which would be considerable.
That increase of one in classification level viewed from Earth would be quite significant. 3% more mass corresponds to about the 4th power of that or 12.5% more luminosity. Earth would be very different if the Sun was 12.5% more luminous. I would think that would be enough to turn Earth at least into a steam-bath, perhaps a full blown runaway greenhouse effect due to the abundance of water and feedback mechanisms. Adding mass would also change Earth's orbit, drawing it in slightly and shortening the year, so the increase from Earth would be a bit more than 12.5%.
The stellar mass math goes something like this.
Lifespan decreases by about the 3rd power of an increase in mass.
Energy output or luminosity increases by about the 4th power of the mass, around 1 solar mass. This increase slows down to about the 3.5th power with more massive main sequence stars.
Surface temperature increases slightly with mass, perhaps 10-20 K per 1% increase in mass, or about a 1% increase in temperature for every 3% increase in mass would be about ballpark.
Radius increases with added mass too, so the heavier sun would be both brighter and larger. The radius increases enough that the surface gravity decreases. A 3% increase in mass corresponds to around a 2% increase in radius, so the size wouldn't appear very different on these scales.
If the Sun became just 0.5% to 1% more luminous (about 1.5-3 Jupiters in mass) that would be in the ballpark to man made climate change so far. Enough to melt surface ice over time and cause oceans to rise, so, significant depends a lot on point of view.
These numbers should be taken with a grain of salt because other factors like metallicity and age matter too. Stars like our Sun also change in size and luminosity during their lifetime, growing both larger and more luminous over time. There are more massive main sequence stars than our Sun that are less luminous because they're younger and the reverse as well, though off hand, I can't name any.
Errors fixed. Thankyou for pointing out.
I want to add, since you mention XKCD in the question, if Randall Munroe was answering this question, he would take it to the extreme, answering what happens if you add million Jupiters, or a billion. Fun times. Without his clever diagrams, it wouldn't be the same, so I'll stop here.