عموماـ يؤتي فلان فلانا حقه
to give someone his due

http://www.thefreedictionary.com/pay dues
to give or allow a person what is deserved or right




اللهم لا تجعل الدنيا أكبر همنا ولا مبلغ علمنا
وأتي كل ذي حق حقه
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خصوصا--في السياق المذكور أعلاه، الذي يؤتي هو الله


فاعطاء كل ذي حق حقه هو صفة محمودة في الإسلام يعني العدالة/ العدل بين الناس مهما كان صاحب الحق مسلما أو غير مسلم
على هذا الأساس، ترجمة
to give the devil his due
غير صحيحة حيث بأنها متعلقة بأصحاب الشر فقط
give the devil her due
Fig. to give your foe proper credit (for something). (This usually refers to a person who has been evil-like the devil.)
http://idioms.thefreedictionary.com/give the devil his due


some examples of this usage "to give everyone his due" in some islamic contexts :

1.
The sole purpose of sending the prophets was to establish Justice in the world and end injustice. Broadly speaking, doing justice means giving everyone his due.

source: http://www.soundvision.com/info/peace/justice.asp

2.
This does not mean that Islam abolishes individual ownership and private business. In fact, it sanctions and respects them in terms of giving everyone his due.

source:http://www.wisdom.edu.ph/Introduction/economic.htm

3.
This is haraam and is not permissible because this will is a will that gives something to an heir, and willing something to an heir is haraam, because the Prophet (peace and blessings of Allaah be upon him) said: “Allaah has given everyone his due, so there is no will for an heir.” (This version narrated by Abu Dawood, 3565; al-Tirmidhi, 2/16 and others.

source:http://islamqa.info/en/ref/22169

4.
to give each individual his due and to regulate his relations with society in terms of value and welfare.
http://muslimcanada.org/social.htm


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Note added at 2 days13 hrs (2013-01-29 16:09:49 GMT)
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why we do not use "grant", but use "give" ?
That is because these two words are not exactly the same in meanings.

Grant ----
means that the thing being given is asked by the one who is receiving the grant. For example, someone applies for a visa entering a country, and he is granted that visa.

or means that the thing being given is from a higher authority or rank to someone with lower rank/authority, without the thing granted being asked for.
for example: Allah granted us a healthy body since we were born.
this kind of things granted are given as a gift /favour.

we could say "we give our parents their due", but I don't think we can say "we grant our parents their due", simply because they have higher authorities than their children, they can grant their children something, but not the other way round.

So depends on the context, if the giver is Allah, then we can use "grant" or "give",because no one can challenge his authority, he is the highest.

and if the giver is human, then "grant" can not be always used as a synonym for "give", as explained above, the difference between them in meanings.

Conclusion:
As a translation, which we can use it for all context in general, I have chosen "give" for "أتى أو أعطى “ as a translation, rather than "grant".