This trio of words — inspired by the Summer Solstice — are completely unrelated historically, but their phonosemantics are remarkably similar.
Sun derives from Proto Indo European swen or suwen, a slightly modified version of the base form saewel, which meant both “sun” and “to shine”. Old English sunne was a feminine noun, and originally all references to the sun assumed that it was female (as in Tolkien’s Middle-Earth — and you may be sure that this was something Tolkien was quite aware of). The sun only became male in English in the 1500’s, long after the noun itself no longer had gender. Phonosemantically sun indicates powerful directed energy (”s”), narrowing toward a goal (”n”), but nevertheless suffused with relaxed, thoughtful qualities (short “u”). Perhaps this reflects the paradoxical power of the sun to both bake you in its heat and lull you to sleep on a golden afternoon.
Summer goes back to Proto Indo European sem, which meant “summer”, and has cognates in dozens of other Indo European languages, from Avestan hama to Old Irish sam (found for example in the name of the festival Samhain, “summer’s end”). If PIE sem is related to PIE swen or suwen (see sun above), there is no way of knowing that now. The sounds of summer begin the same as sun – the “su” indicating strong directed energy that is nevertheless relaxed and roomy — but the result is a manifestation of tremendous undirected power (”mmer” — and you better believe the nickname “hummer” reflects the same intuition).
Summit, meanwhile, is related to the Latin preposition super, meaning “over” (and borrowed into English as a very useful prefix with approximately the same meaning). Super itself comes from the Proto Indo European preposition uper, which had the same meaning, and is also the ancestor of the Greek hyper. The Latin preposition super was the basis for the Latin noun summum, “the highest”, which was clipped in Old French to become som, “hilltop”; the diminutive of this was sommette (”little hilltop”). Finally, this French word was borrowed into English as summit in the mid-1400’s. When it was borrowed, it doubtless only meant “little hilltop”, but over time its meaning has expanded again and now approximates the original Latin summum again. Phonosemantically, it is almost the same as summer – strong, directed, relaxed energy leading to a manifestation — but this time the manifestation is not undirected energy, but a definite upward, tense motion along a path (”it”), like a trail to the top of a mountain.
Finally, a bonus word: Solstice is related to sun – it derives directly from the PIE root saewel mentioned above, via Latin. Here “sol”, the sun, is directed, wholesome energy, filling all available space; and “stice”, from a Latin suffix meaning “standstill”, channels that energy into a tight, high place.
Joyous Solstice to you and your loved ones!