First of all: the genetic makeup part of a DNA test result is fun, but isn't something that can be relied upon or ever provide 'proof', because it's based on the DNA reference groups and samples held by the company that made it. It's very much broad brush strokes.
For tracing ancestors, the valuable evidence is obviously the DNA itself.
It's very unlikely, but not impossible, that you might be able to trace an ancestor who was alive in 1588 via autosomal DNA testing. This is due to two problems:
- 1588 is so long ago that your mother's DNA is unlikely to retain any substantial segments from this ancestor. Even if the story is true, due to the random apportioning of DNA in each generation, your mother might well not have any genetic trace of this ancestor (and by extension, only a tiny proportion of this person's other ancestors are likely to carry any traceable segments, and those who do may well not have had their DNA tested).
- Very few people's trees go back that far. To have a real chance of tracing an ancestor back that far, you'd have to hope that other ancestors of this person had phenomenal family trees. In practice very few people have family trees that reach back to 1588.
But you know what? I'd give it a go anyway. You're unlikely to ever prove it, but you're very likely to find out other new and interesting things on the way.
I'd suggest uploading the DNA results to ftDNA, Gedmatch and MyHeritage in order to be in as many databases as possible (you might also consider testing with whichever out of 23andme and Ancestry your mother didn't test with)
If you're (unbelievably) lucky, you might be able to identify a cluster of matches who all seem to have the same common ancestor, and that ancestor might be the one you're looking for. I've come across cases of such an ancestor born in the 1640s/1650s, but you're going back another hundred years, which lessens the chance.
But I'd say since you've started your DNA journey, keep going!