When listening to native Irish speakers, you may notice that some of them say ‘héin’ instead of ‘féin’ when referring to themselves, for example, mé héin / liom héinig. By the way, this héin should not be confused with the plural of birds e.g. d’imigh na héin, just in case you were thinking that way.
I digress. In a nutshell, féin and héin mean exactly the same thing and spelling-wise the latter should always be héin. People keep spelling it in the Irish language as fhéin for no good reason but this is not correct and misleading. It should never be spelt fhéin or fhéinig as the letters “fh” do not represent a “h” sound in Irish. Rather they represent no sound at all. The word would be pronounced with no initial consonant at all. If you want to represent the pronunciation it would be [‘he:nʲ].
N.B. The spelling fhéin is a remnant of pre-standard spellings and importantly fhèin is still used in Scottish Gaelic e.g. Bidh agad ri a dhèanamh thu fhèin – You’ll have to do it yourself.
Many native Irish language speakers will still write féin but say héin. Further evidence of héin being used is in the Linguistic Atlas and Survey of Irish Dialects (LASID) pg. 297 which shows the full map of Ireland. Below is a snippet, but you can find the digitised version here.

But why is it spelled féin if it’s pronounced héin? Historically it was pronounced with an f. It was still pronounced with /f´/ in some places, well was, in the sense as long as there were native speakers there and was heard during the 20th century in some Munster speakers at least. Féin exists alongside héin in at least West Munster.
The word féin is generally pronounced héin after a vowel and is always pronounced this way in Ulster Irish. In Munster, the form féinig (or héinig) are also just as common. Alongside féinig (héinig), there’s also another form: féineach (héineach) in Munster.
A side note: both céin and féin (and variants: fadéin) existed in Old Irish. An example below is from Pangur Bán, specifically verses 1 and 4.
First Verse
Messe ocus Pangur bán,
cechtar nathar fria saindán;
bíth a menma-sam fri seilgg,
mu menma céin im saincheirdd
Fouth Verse
Gnáth-huaraibh ar greassaib gal
glenaid luch ina lín-sam;
os me, du-fuit im lín chéin
dliged ndoraid cu ndronchéill.
One other thing: péin / péinig
If you delve further into the Irish language you will also see another variation péinig / péin. Such forms were also found in Clare and are from a reanalysis of sibh féin. Thus sibh féin became si péin which was reanalysed as sib héin, leading to sib and thus lib, dhaoib, etc.
You see you’d get devoicing of the b in sib from the “h” in héin, so it would sound like si péin and this is why it was interpreted that way. Clare made even more extensive use of these forms with the likes of oraib instead of the Conamara oraí, rather than being limited to monosyllables. The form is also seen in places like Muscraí and Corca Dhuibhne.
But also worth noting that péin/péinig is only when with the relevant labial positions e.g. dom péinig, due to the labial. Péin usually follows M e.g liom péin.
Examples of this in literature include Feis Na NGleann: A Century of Gaelic Culture in the Antrim Glens: Nach tuirseach mise anseo liom péin, Is ioma’ Nollaig a bhí agam péin, Is chan aithnim péin an Domhnach.

One further thing!
Interesting to note: There are other places where f changed its pronunciation to h, for example in most future tense and conditional endings: déanfaidh /d´ianhəɡ´, d´eːnhə/ (déanha(ig)).
But also note that in some forms like dhéanfá ‘you would do’ it’s actually f: /ɣ´ianfɑː/ (which is pronounced as written) or déanfar ‘(it) will be done, one will do’ /d´ianfər, d´eːnfər/. Basically, déanfaidh is said as if déanha(ig) (and dhéanfá and déanfar are said as written.

GRM…
I love the internet…as a Gaidhlig learner watching politics it caught my attention…and you are here to provide the answer.
Fingers crossed I can engage in Gaidhlig soon and Gaeilge eventually