There's multiple factors to consider, and you may have to do some more world-building to find out the proper outcome.
Determining intermediary language(s)
First, 10th-century Arabs were not the neighbours of 10th-century Englishmen. A word would not have gone from Arabic straight to English: the two people were barely in contact. The English would have first heard of this new world not from Arabic sources, but from people who talked to people who talked to Arabs. One or several intermediary languages need to be defined.
For "Gibraltar", that was Spanish (they coincidentally also spelled it Gibraltar). A different example, "Algebra" comes from the Arabic "al-jabr" (الْجَبْر), via the Medieval Latin "algebrāica", which was the lingua franca at the time among scholars and theologians. With algebra being first introduced by a science paper, it makes sense that Latin would be the intermediary language.
Take a look at this Wiktionary page for more examples. Often there can be multiple intermediary languages. "mummy" (the Egyptian embalmed corpse, not the British word for mother), came from the Arabic "mūmiyāʾ" (originally from the Persian "mum", meaning wax), through Medieval Latin "mumia", then through Middle French "momie", and then William the Conqueror took the Normandic dialectal version of the word, "mumie", with him when he invaded England and made Anglo-Norman the language of the nobility.
"Magazine" had two intermediary languages: Arabic "maḵāzin" -> Italian "magazzino" -> Middle French "magasin".
What you have is that each intermediary language modified the word to suit their inventory of sounds, their phonology. Then they may have added suffixes or prefixes, pluralised it, put it in a different grammatical mood or tense, before it moved on to the next language.
And then both those words went through all the vowel and consonant changes that turned Middle English into Modern English - which is the next step.
So, the first thing to figure out is how the word would have spread to English. Who did they learn the existence of this world from? And how fast did the word travel - literally?
This is something I cannot really make assumptions about without knowing more of your world, what states did and did not exist in your version of the 10th century and whether the Arabs kept it a secret, or told everyone about it, or just their trading partners and allies, et cetera; so I'll finish the next part of this answer when you fill me in on that :)
Alrighty, we now have more to go on!
The ''Door" opened in Sijilmassa, a silk road trading hub connecting North Africa to subsaharan africa. News quickly traveled to Andalusia of jinn emerging from the unseen world, building strange edifices and doing fantastic magic. From Andalusia, the stories spread to the major cities of Castille, by the time Sijilmassa fell to these strangers (circa 935AD) and a stories of a terrible plague, observed by a Coptic Monk named Baba Boutros and inscribed in latin correspondences to the church in Rome and Alexandria
@AlexP made a great proposal for an Italian/Latin-based path; I'll try to go with the Iberian route. The way I am proposing is Morocco -> Andalusia -> León (Castille did not exist yet) -> England. I'm assuming a naval transfer of the word from Iberia to Britannia; since it would be rapidly spreading news, seafaring traders would have carried it directly. Also, both León and England were thriving kingdoms at the time, while France was weak and decentralised and Germany in the middle of a succession crisis. I think word would have spread directly; given the significance of the event.
Following the road
Morocco -> Andalusia
First off, I am going to assume that no bastardisation occurs in the travel between Morocco and Andalusia. That is because both regions speak Arabic, and would directly understand each other. You would have a calque rather than a loanword: the roots are recognised, therefore the same roots are used for the version of the word native to the new speaker's region. Perhaps different, equivalent roots would be more idiomatic, but I do not know enough of medieval Arabic to know that for sure, so I'm going to keep them the same.
Then it becomes the task of figuring out which term you would end up with in Al-Andalus. There are two languages in the region to choose from: Mozarabic and Andalusian Arabic. The former is a blend between the tongue of the Romance peoples left behind by the Roman Empire and the Arabic that the conquerors imparted on them in the past two centuries: the latter is what the Arabic speakers there spoke in informal settings, influenced a bit by the local Romance languages, and separation from other Arabs by distance. Classical Arabic was still used for formal communication, e.g. liturgy.
I have decided on the latter. I think the people with the most means to travel distances and spread the news would have spoken Arabic, rather than the dialect blend found in the lower class. However, I am assuming they would not use standard Arabic to communicate with the Leonese: the two peoples had been neighbours for centuries by that point; I am sure translators would be better acquainted with the local variety than with the Caliph's Arabic. Andalusian Arabic it is.
What features did that dialect have? They are described in contrast to Standard Arabic, so I am going to start with the phonetical representation of عالم الغيب (the version I'll be working with because it's the only one I have in Arabic writing), and apply changes I can find to the best of my abilities.
My guess for the IPA is /ʕaːlim alɣajbu/ (based on ear and the transliteration of the word)
So, what do we know about Andalusian Arabic phonology? The wiki article gives us some useful info:
The vowel system was subject to a heavy amount of fronting and raising, a phenomenon known as imāla, causing /a(ː)/ to be raised, probably to [ɛ] or [e] and, particularly with short vowels, [ɪ] in certain circumstances, particularly when i-mutation was possible.
Alrighty, you really need to know more of the Arabic language to understand what those "certain circumstances" entail. But since we've got two more language transitions to go, I'm going to be a bit lazy here and turn every non-diphthong /a/ into an /ɛ/.
Contact with native Romance speakers led to the introduction of the phonemes /p/, /ɡ/ and, possibly, the affricate /tʃ/ from borrowed words.
Great, great, it doesn't tell me where the /g/ is introduced... I am really tempted to just replace the rhotic [gh] (/ɣ/) with a /g/, but that would leave me without anything interesting to solve in the next transition :) So I'll leave it alone for now.
Monophthongization led to the disappearance of certain diphthongs such as /aw/ and /aj/ which were leveled to /oː/ and /eː/
Oooh, great. We can make use of that for our diphthong!
So, the end result is /ʕɛːlim ɛlɣeːbu/. Next stop, the Leonese border!
Andalusia -> León
I am starting to get really lazy here, but this is the best of my abilities: I'm just going to check phoneme by phoneme whether there's a direct equivalent, and if not, what that phoneme has been transliterated to historically.
The Kingdom of León spoke... five languages. Astur-Leonese, Latin, Castilian, our good friend Mozarabic, and Galician-Portuguese. Awesome. Let's go with Leonese; we haven't yet ran out of extinct dialects to play with.
The /ʕ/ doesn't exist in any Iberian language, and from the examples it looks like it just gets ignored.
Spanish dialects are quite short on vowels, and they don't have the /ɛ/: except for Andalusian/Murcian dialects (who probably got them from Arabic), but those did not exist yet in the 10th century. /ɛ/ in those dialects corresponds to /e/ in the standard language (and Leonese), so that replacement is easily made.
Nothing special about /l/ on its own. /i/ and /m/ also stay the same. Now we get to our fancy consonant, the /ɣ/ - which is a phoneme Spanish possesses: but it gets turned into a voiced velar stop following a nasal consonant, such as the /l/. So where Andalusian Arabic failed, Leonese makes the exact same transition. /ɣ/ becomes /g/.
Spaniards are no stranger to the /b/, and neither is Leonese. Now, the final /u/ is a bit of a problem: Spanish words typically do not end in closed vowels like the /u/. The u was never stressed in the original Arabic, so maybe it can be left out... wait, Leonese phonology to the rescue! The only open vowel allowed to be unstressed is the /a/, so that's another quick replacement to make..
/eːlim elgeːba/ Next stop: England!
León -> England
Here there's a new issue. Old English had an incredibly rich sound inventory: every phoneme in /eːlim elgeːba/ can be found there as well. So my previous method would produce no changes at all. Let's try the other tactic: finding example words...
This list is not entirely helpful. Many of these words barely changed at all. Salsa from salsa, embargo from embargo, guerrilla from guerrilla, canyon from cañon. Many of them are also more recent loans than the time period we had been speaking of: Spanish became important and influential on English only after the discovery of the New World and its colonial empire.
Words with bigger changes do seem to follow a few tendencies. For one, (Old) English is more tolerant of consonant clusters, and unaccented vowels occasionally drop out. Example: crimson, from cremesín (though that one passed through Latin too); quadroon from cuarterón; cockroach from cucaracha. This is not very scientific, but I am going to swallow the /i/ and make the result more alliterative as well (also happened to avocado from aguacate); /elmelgeːba/. The lm-cluster would probably velarise the /l/ to a /ɫ/, as we see in words like "palm".
With the ending you can do whatever you want. I tried a couple things, but this close to the finish line I feel shy from settling on something. You could transform it to a diphthong, after potato from patata, and get something like elmelgeob. Just looking at endings of old English words, -pe is a lot more common than -be or -ba or anything like it, so maybe elmelgape does the trick?
None of this is as scientific as the linguistic terms I throw around make it sound. Many transitions are based on random chance. Someone with a lisp could have been the messenger to the English court, and that could have shaped the pronunciation of the word forever.
I think this answer should mostly serve as an illustration for the process of the bastardisation of a word. Taking the word and making it fit the next language's phonology is sort of like the telephone game concept in the other answer, but I richly interpreted every step along the way.
So I hope that this idea can help you make your own version to suit your needs!