"Teagan" is an Anglicization of the Irish-Gaelic name "Tadhgán." "Tadhgán" is a diminutive form of the more common name "Tadhg." The Irish-Gaelic diminutive suffixes '-án' & '-ín' are roughly equivocal to the '-y' ending in English. A diminutive suffix indicates something little but also something dear and affectionate.
It has long been definatively thought the meaning of "Tadhg" is 'poet,' and many still rightfully hold to this theory. But someone on another message board pointed out a whole other theory, not necessarily new but new to me. "Tadhg" may very well mean 'badger.' For background information, see the second link below which is etymologically sound.
Both of these definition categories are traditionally likely. Occupations that are also social positions like a 'poet' (see Riordan/Ríoghbhardán) and animal-based names (see Matháin/Mathghamhain, Conán, Rónán, Oisín, and countless others) are well established commodities.
The definition of 'little poet' or even 'little badger' is not exactly correct. Though "Tadhgán" does have a diminutive suffix, this would not correlate to its definition. It is a diminutive suffix applied to an existing name. By comparison, "Thomas" means 'twin.' "Tom/Thom" and "Tommy" do not mean 'little twin.' Conversely, the Irish name "Lonán" (same '-án' suffix) does mean 'little blackbird' because "lon" is not a name in and of itself.
I can state definatively though, the 'doe/deer' and 'beautiful' definitions of "Tadhg" are wholely without merit. I cannot reason how these associations were made.
Regarding spelling variations, the primary Anglicization of "Tadhgán" as a given name is probably "Teagan," but the surname of the same derivation uses the alternative spellings "Teegan" and "Tagan" as well. The root name "Tadhg" has countless Anglicized variations: "Taig," "Teague, "Tigue," "Teigue," "Teige," "Tighe" and possibly more. There is also another Gaelic diminutive form, "Taidhgín." This name shares the Anglicized forms of "Tadhgán," but sometimes spells them slightly differently.
Note that American-English mostly pronounces "Teagan" as [TEEG-uhn] and Irish-English/Hiberno-English mostly as [TAYG-uhn]. And just for the sake of completeness, I should say in Irish-Gaelic "Tadhg" is pronounced [TIEG] (like the first syllable of 'tiger'), "Tadhgán" is [TIEG-awn] (a long 'á' as in 'lawn'), and "Taidhgín" is [TIEG-een] or [TIEG-yeen].