Measuring the Shape of the Earth
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"What point on Earth is the farthest from the Earth's center?"
If it's being asked at all, you know it's probably not Everest, and indeed it's not:
https://en.wikipedia.org/wiki/Summits_farthest_from_the_Eart...
A good theory for the solid earth tide only arrived in first dozen years of the 20th century and was only practically measured with the arrival decades later of very long baseline interferometry (up to extragalactic radio sources), satellite (and lunar) laser ranging, geodetic GNSS, and other tools in modern geodesy.
The solid earth tide has a maximum of a few tens of centimetres, not metres. So even in the 21st century, solid Earth tides, being small and difficult to measure without specialist equipment, are for most practical purposes completely negligible.
Your astrophysicist relative may be interested in some details.
At <https://geodesyworld.github.io/SOFTS/solid.htm> there is a calculator (and notes) giving more information than a lay person is ever likely to want to know ("Fine print: This section is not for the faint of heart."); there is a reimplementation and modification of solid in R at <https://rdrr.io/cran/solidearthtide/>; there is the ESA page with useful simplified formulae and graphs at <https://gssc.esa.int/navipedia/index.php/Solid_Tides> since solid earth tides can be relevant for astronomy; or alternatively the wikipedia page for https://en.wikipedia.org/wiki/Earth_tide is decent and more accessible, and has a link to tidal Love numbers at the bottom.
Your relative can look up the fluid and tidal Love numbers for main sequence stars (and how dynamics and dissipation works with fast rotation), and there is literature for relativistic Love numbers for white dwarfs and neutron stars and so forth (although nonspinning black holes have none: Binnington & Poisson "Relativistic theory of tidal Love numbers", <https://arxiv.org/abs/0906.1366>), and should have little trouble following methods for arriving at tidal Love numbers given different equations of state for visecoelastically differentiated bodies.
Earth has https://en.wikipedia.org/wiki/Planetary_differentiation and when you dig into solid Earth tides in detail you get to confront all sorts of unknowns. In practice, it is easier to start with the precise measurement of rotation of a real body e.g. Earth, Saturn, Io, super-Earth exoplanets (a nice example is at <https://doi.org/10.1051/0004-6361/201731775>) and work backwards to the interior structure. This is perhaps less true for the Earth since we can use time-of-travel seismology and gravimetry and have ready access to rock samples.
ETA: (in part so that I don't lose the reference), this astronomy stackexchange summary of the tidal Love numbers is extremely well written: https://astronomy.stackexchange.com/posts/46889/revisions (and I'm sure your astrophysicist relative would appreciate it too).
https://en.wikipedia.org/wiki/Tide-Predicting_Machine_No._2 (be sure to see the photos).
Your comment about the solid earth tide was pretty neat because few people who aren't professionally into geodesy or the study of Earth's gravitation think about that.
As I mentioned, there are tides in all sorts of bodies elsewhere in our solar system (and beyond). The ones people think about are atmospheric tides and ocean tides (within e.g. Europa), or how it couples to volcanism (e.g. Io's magma and Enceladus's ice).
However, scattered about our solar system there are "asteroids" in pairs, where one is what amounts to a blob of kitty litter to large gravel, and the other is a more solid body. The rubble-pile is especially interesting in a tidal context. I just submitted https://news.ycombinator.com/item?id=39561656 which the Royal Astronomical Society announced yesterday in various places including <https://twitter.com/RAS_Journals/status/1763219322732786054> (the image is worth a click, but also appears as Fig. 1 in the full article at the HN submission; if you hate X and/or extra clicks, this is a direct link: <https://academic.oup.com/view-large/figure/441211342/stae325...>).
When general relativity students learn about decompositions of the Riemann curvature tensor they are sometimes taken to the idea of a sort of cloud of coffee-grounds (with the special provision that they pass through each other, always have gaps between them, or are otherwise non-collisional and therefore non-clumping; and they have no internal pressure) to see a variety of ways in which one can find "tidal forces" encoded in e.g. the Weyl curvature tensor (which is extracted from the Riemann curvature tensor). Baez is an example of this <https://math.ucr.edu/home/baez/gr/outline2.html> and <https://math.ucr.edu/home/baez/gr/ricci.weyl.html>, although textbooks generally stick to "test particles".
Rubble pile asteroids, while essentially not needing any post-Newtonian gravitational corrections at all, are a neat step up in difficulty because they do clump. But unlike larger bodies, they aren't fused by e.g. melting; the rubble can shift around unlike a fused rock or icy concretion; and internal pressures and shear stresses are low. So the rubble (how it shifts around, what shape it tends to adopt) will be affected by tides induced by the sun and possibly other solar system bodies closer to them.
Personally I like the idea of depositing a crash test dummy loaded with sensors onto the surface of (probable) rubble-pile 101955 Bennu. Is it like a ball pit, or like a bean-bag chair? Does the dummy sink or get buried in landslides or fountains of material? (We already know we can send things there (and return them): <https://www.nasa.gov/news-release/nasas-bennu-asteroid-sam>).