My father had a Ph.D. in mathematics. He was a consummate mathematician and enjoyed reading about all topics related to math. He passed away a couple of years ago. I have yet to find a suitable place to donate the many theoretical math books he had in his library. I want these books to get a second life by being read by those who would understand and value their content. (I could provide a list of titles if that would be helpful.) Any guidance from this forum would be greatly appreciated. Thank you.
Most large universities have a library that could take them, and where students could find them and "give them a second life." If the library doesn't want them, many math departments have a common room with a miniature library inside. You could ask the chair of the department (or the department's administrative assistant) if that common room has space for more books. Again, this way they get to a place where mathematicians and math students can find them.
As already pointed out by Stopple in the comments, AIM is another option, but then you might incur shipping costs. Another option, mentioned in the thread What to do with antique math books? linked by Carlo Beenakker, is the Internet Archive which might want them, to scan them to put them online. But, again, they might not cover shipping. For details, see: How do I make a physical donation to the Internet Archive?.
I think focusing on your nearest large university is the way to go.
The London Mathematical Society has a programme "Mentoring African Research in Mathematics". The main people involved are listed at https://www.lms.ac.uk/MARMBoard. In particular, the chair is Frank Neumann (https://le.ac.uk/people/frank-neumann). I suspect that they know of many places that would welcome books but have few resources to buy them. Of course, someone would have to pay for shipping, which might or might not be an issue.
In Brazil, people are used to do as most people here suggested: find a public or university library that could receive your dad's collection. It would be interesting observing a trade-off, since you may want the collection to be as well care as possible, while having the maximum impact on the community it will serve. A public university library would be the primary option here, specially those that are more inclusive and/or suffer from a lack of resources to assemble their collection.
I too live in the greater Phoenix area and as a member of the Humanist Society of Greater Phoenix (official building is in Mesa), I'm sure they'd happily take them off your hands. There is a library at the center as well as the option to donate to the "Little Free Library" they host.