Well, Tar seems to me like it would relate to "ta" which generally shows that a word relates to the patten. See: ta'veren, ta'maral'ailen, etc. I would also extrapolate that Tarmon Gai'don does not simply mean "Last Battle" because we know that gaidin means brother to battle, so changing a single vowel likely would not regress the word to its root in "battle."

I'll come back to Tarmon Gai'don in a moment.

Back to Tar Valon... because we see "val" involved in a few words (valere, valdar, etc.) I think it does likely relate to "guard" as is suggested by the old tongue dictionaries online. I would, however, suggest that it also implies greatness, heroics, a presence. Especially because we see the "val" prefix on the Horn of Valere and on the Heart Guard of Manetheren... two groups that are known in history for their heroism.

So, I would think that Tar Valon likely has implications of standing to guard the Pattern itself, despite whatever comes against them. It could also hold implications of "last guard" due to the association with tarmon gai'don... I doubt that implication is literal, but would have been in the minds of the women founding the island city.

Now, back to the last battle... I would suggest that "gai" does simply mean "battle" with "don" carrying the "final" connotation... so, if the name really just meant "last battle" then, by my understanding, "gai'don" would be sufficient. But we know it is not.

Above, I suggest that "tar" means it relates to the pattern, so "tarmon" should also relate to the pattern, with the "mon" suffix stressing the importance and dire situation.

So, I think that "Tarmon Gai'don" should mean "The Final Battle for the fate of the Pattern"