The word "spondee" is a bit unclear when talking about English. There are certain disyllabic words that could be said to have a major stress on both syllables: words like the -teen numbers (e.g. nineteen) and two-letter acronyms (e.g CD) where the primary stress or accent falls on the second syllable when the words are spoken in isolation, but when the word occurs in a phrase, like nineteen men, the accent may fall on the first syllable (apparently, to avoid having two accented syllables in a row). I somewhat doubt that there is any pronunciation of "Peyton" that actually has an accented second syllable in this sense.
The other thing "spondee" could refer to is words that have only one major stress, on the first syllable, but that also have "minor" or "tertiary" stress on the second syllable. Many disyllabic compound words fall into this category, like outcast (n) or household.
I wonder if you are saying that your nephew pronounces the second syllable of Peyton as an "accented" syllable when you really mean that it sounds to you like it has a "minor" or "tertiary" stress. For comparison, the word "latex" is generally pronounced with an unreduced "short e" sound in the second syllable, which some American English speakers hear as a kind of stress (see MW's transcription "\ ˈlā-ˌteks \") but it's clear that the primary stress or accent falls on the first syllable. The two syllables are not equally accented. Likewise, I would expect that your nephew's pronunciation of "Peyton" actually has only one primary stress, and that it falls on the first syllable, but perhaps the vowel sound used in the second syllable makes it sound more like a stressed syllable to you. (However, the fact that you nephew realizes /t/ in this context as a glottal stop rather than as an aspirated plosive suggests that the syllable is not being treated as stressed in his phonological system.)
You write your pronunciation as "PAY’-n", which to me suggests that you actually pronounce this name with a syllabic nasal consonant [n̩] as the nucleus of the second syllable.
Unfortunately, I don't know of linguistic literature or studies that mention this, but I have seen anecdotal accounts suggesting that the realization of /ən/ after /t/ or /d/ as a syllabic nasal [n̩] with no preceding phonetic vowel sound may be more common for older American English speakers than it is for younger speakers. Older speakers remarking on this apparent phenomenon seem to vary in whether they describe the vowel that they hear younger speakers using before [n] as sounding more like [ɛ], [ɪ] or [ǝ].
For example, this blog post, "Desyllabication of /n/ in Consonant Clusters", by Michael Shapiro:
American English in the last decade or more has manifested a phonetic change whereby what was previously a syllabic /n/ in the clusters /dnt/ and /tnt/ at the end of words has instead developed an epenthetic [ɛ] preceding it. Accordingly, whereas the older normative pronunciation of words like student, hadn’t, didn’t, and patent typically had no vowel before [n], now the younger generation of speakers inserts an unstressed open mid-vowel [ɛ] before it.
As well as the questioner (and some commenters) in this Language Log post: Ask Language Log: Trend in the pronunciation of Clinton?
From David Russinoff:
I wonder if you've done, or are aware of, any research relevant to the following observation. In the articulation of a "d" or "t" followed by a schwa, the tongue may or may not leave the alveolar ridge. (I just did some cursory research on parts of the mouth and hope I got that right.) My (highly unscientific) observation over recent years is that, at least in the pronunciation of certain words, such as "student", removal of the tongue is increasingly common. In fact, this trend is so apparent to me that I find it remarkable that most people don't seem to have noticed it. I also have an impression that the trend is especially pronounced (unfortunate choice of words) among younger speakers, but my attempt to support this observation by listening to various pronunciations of "Clinton" over the past four nights failed miserably.
To check if the same difference exists after /d/, maybe you could listen to how your nephew pronounces words like student, garden, or hidden.