The 'T' and the 'H'.
The major division of pronunciation divides into those who pronounce the 'T' and 'H' together as in 'thumb' and those who pronounce the 'T' separately. The origin of the name with 'Groit' and 'Ham' suggest clearly that they originally were pronounced separately. It is only when it is written down that others can become confused. Perhaps this is why most American Greethams pronounce their name with the 'T' and 'H' making a 'TH'. American English tended to taught with the ridged phonetic system invented by Mr Webster in the 19th century. Each part of a word was broken up into it's component parts and pronounced separately. Teachers would automatically see the 'TH' as always making a 'th' sound, thus children were forced to pronounce it that way. To my knowledge every American Greetham family pronounces it that way.
Before the 19th century most Greethams in the UK wouldn't have a clue how their name was spelt. On those occasions when their name was written down, perhaps in a parish register, it was up to the vicar or curate to write the name out best they could. Many of the clergy would be aware of the spelling of the village in Lincolnshire and would spell it correctly, others would spell it as they thought they heard it.
When the 'E' is prounced 'ehh', then 'GRETHAM' is an obvious spelling. In many cased the 'H' is missed out in the pronunciation leading to 'GRETAM'.
Another shift in the vowel leads to the 'A' being pronounced as a 'U', thus I have come across, GREETUM', 'GRETUM' and even 'GHRETUM'.
The first set of vowels can lead to some classic spellings and pronunciations. A longer 'eh' sound can lead to 'GREER-TAM' or with an 'ay' sound simply 'GREAT-HAM'. In some parts of the UK long 'ee' sounds are avoided and that can produce a sharp 'eh' or even an 'i' sound. Thus you will find 'GRET-UM' and 'GRITTAM'.
I have come also across a cockney 'GREIFAM', a Lancastrian 'GREET-HUM', and a Dutch 'GRETAAM'.
Perhaps the most unusual form is from a Greetham who lives in Japan where it is neccasary to conform to their somewhat restricted pronunciation sounds, and is reproduced using the Japanese symbols . . . 
and is rendered 'GU-RII-TA-MU'.
Can you add to my list or have interesting ways 'Greetham' has been spelt?
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In an alphabetical list 'Greetham' is always between the Greenwoods and the Gregorys. We are fortunate that if the 'H' is missed out then it will still be in the same place. To be on the safe side you will need to check GREATHAM which occurs earlier and GRETHAM which occurs later.
In a computer database or website search, input GRE*T*
(The '*' is a wildcard and will allow seaches for occurances of any letter(s) or none at those positions. This will cover most variations.)
If you still can't find it then try, GRIT*
If a name still doesn't come up then you may have to input GRE* or GRI* In these cases try to fill in other fields because it will generate a lot of names otherwise.
