"Naïve" yet "naivety"?

There isn’t really any standard for the use of the trema in English, so I don’t think there is any deep “why” to the behavior of this spell-checker. As mentioned in the comments, “naivety” has an anglicized spelling of the suffix, so it’s not the same as the French spelling even if you use ï. In contrast, the form naïve does exist, ï and all, in French—although unlike in English, it’s only used as the feminine form of the word. (The French masculine form naïf can sometimes be seen in English texts as a noun, but I don’t think it’s ever used in English as an adjective.)

The difference could arise because naïve is a direct loan word from French, but naivety is an English word adapted from the original French naïveté.

A quick search for naivety and naivete (without diacritics) in Google Books yields strikingly different Ngram graphs for British English and for American English. Here is the Ngram chart for naivety (blue line) versus naivete (red line) in British publications for the period 1800–2005:

And here is the corresponding Ngram chart for naivety (blue line) versus naivete (red line) in U.S. publications for the same period:

The first chart indicates that the rise of the spelling naivety in British orthography is a fairly recent development—one that began to take hold only in the past 100 years. In U.S. orthography, in contrast, the spelling naivete has always been considerably more common.

This difference in preference is worth taking into account before you begin to worry about whether and where to place the diacritics.

Hitting it straight off here, naïve is a loan-word (a word that was derived from another language yet has avoided entire english assimilation) yet naivety is an english modification to the word. Changing the word to english rules force the word into a completely english state, removing the dieresis (¨) from over the i. In addition, personal experience leaves me hearing the word naivety as nai-IHV-ity more often than nai-EVE-ity. Although I am unsure, this pronunciation seems to come back to the english appropriation of the foreign word. For those who don’t know, naive requires a dieresis from the diphthong created by “ai” in a single-syllable word, otherwise pronounced like the letter i. In the noun form of naivety, the stress is maintained on the ai. In all measure, one should retain the dieresis in naivety to keep proper pronunciation, but the english refusal of accents letter altercation keeps its speakers in a constant state of confusion in what should be simple.