Since I already have a clear picture of the Jamtlandic orthography, I think I can claim the alphabet in this post. The claim is that the Jamtlandic alphabet is given by:
Minuscule: a á b d ð e ë é f g h i ï í j k l ł
m n o ó p r s t u ú v y ÿ ý æ ø
Majuscule: A Á B D Ð E Ë É F G H I Ï Í J K L Ł
M N O Ó P R S T U Ú V Y Ÿ Ý Æ Ø
This alphabet is basically based on the Old Norse one.
The acute diacritic ( ´ ) denotes vowels coming from old long vowels, denoted with an acute diacritic in Old Norse. The umlaut diacritic ( ¨ ) denotes vowels which do not cause an expected softening of g and k. (The vowels e, i and y cause softening, which the umlaut diacritic stops. This has to do with the vowel levelling phenomenon in Jamtlandic.)
Like in Faroese, the letter ð is silent. It is necessary in order to separate minimal pairs such as kaste (inf.,pres.) vs kasteð (imp., past part.) 'throw', in most Jamtlandic dialects pronounced [kʰa.ast] and [kʰastə], respectively, but in some (archaic) dialects pronounced [kʰastə] for both. A unified orthography for all dialects requires a special symbol to denote the prevention of dropping the ending vowel. The etymology for this prevention is the letter ð, so we use it. The use of ð is generalized, e.g. kné 'knee' vs knéð 'the knee', both pronounced [kneː]. We don't even need minimal pairs: góð [guː] 'good', from Old Norse góðʀ [goːðɻ].
The letter ł was discussed in an earlier post.
The letter æ doesn't come from Old Norse æ (which has coincided with é in the alphabet proposed here), but is a product of the vowel levelling. E.g., Old Norse spila 'play' and tala 'talk' have become Jamtlandic spæla [spælɐ] and tæła [tʰæɽɐ]. (This is actually a concrete near-minimal pair in the motivation of the "thick"-l symbol ł.)
In most cases, the letter ø, [øː] (long) or [œ] (short), comes from Old Norse œ/ǿ, i.e., a long ø [øː]. One rare exception is øks 'axe', from Old Norse øx. (Note that Old Norse ø never comes from an i-umlauted o. In the case of øx, the etymology is perhaps *akwezī.)
I think this concludes a superficial discussion about the proposed alphabet. Of course, later I'll be more detailed.