Praise be to Allaah.
If the name Iram Fatima is commonly used among people who
speak your language and people from your country, and it does not carry
any meaning this is contrary to Islam or its etiquette, then there is
nothing wrong with using this name, because there is no condition
stipulated in sharee‘ah that the name should be mentioned in the Qur’aan
or Sunnah, or that it should be an Arabic name, and there is nothing in
sharee‘ah to indicate that it is preferred for non-Arab nations to use
Arabic names. Rather one should avoid using names that are exclusively
used by followers of other religions and that are usually used by the
followers of other religions, such as George, Peter, John, Matthew and so
on. It is not permissible for the Muslims to use such names because that
is imitating something that is exclusively theirs.
Quoted from Ahkaam Ahl al-Dhimmah by Ibn al-Qayyim,
But if a non-Arabic name has a good meaning, there is nothing
wrong with using it and giving it to one’s child. The Messengers and
Prophets (blessings and peace of Allah be upon them) had good names and gave
good names to their children that were based on their customs and
traditions, and they were not Arabic names. Such names include Isra’eel
(Israel), Ishaaq (Isaac), Moosa (Moses) and Haroon (Aaron).
Imam al-Mawardi (may Allah have mercy on him) mentioned some
matters that are mustahabb with regard to choosing names. Among other things
(The name) should have a good meaning and be appropriate to
the situation of the person to whom it is given; it should be a name that is
usual among the people of his class, religion and standing. End quote.
Naseehat al-Mulook, p. 167
Because this word, Iram, is not used in Arabic we looked for
it in other languages and – from looking on some websites – it became clear
to us that the name Iram is used for females in some Muslim countries and
its meaning in their language is: a garden in Paradise.
If that is in fact what it means in your language, then there
is nothing wrong with using this name; in fact this is a good meaning, but
we did not understand why it should be added to the name Fatima, because
Faatimah was not singled out for Paradise to the exclusion of other women.
So to be on the safe side it is better to choose just one of the two names,
either Fatima or Iram, if that is customary in your country.
And Allah knows best.