The sign 'gi' obviously depicts a plant and its foremost meaning is 'reed' (Labat - Manual d'Epigraphie Akkadienne, 6th). The land is/was largely a reed-swamp. Without a name I would probably call it 'the reed land' or perhaps (referring to the chief or the people being masters of working with reed) something like 'land of the masters/lord(s) of the reed'. But I have not yet come accross such an interpretation. Hayes - A Manual of Sumerian Grammar and texts, 3rd, 2019 p66-67) for example lists some proposed etymologies that, to me, seem very far fetched. So, what is wrong with 'reed'?
There are several signs that are read GI and they have different figures to contrast them. It seems that gir15 is the Sumerian word for saying "Sumerian".
Eme-gir15 = Sumerian language, also written eme-gi-ra, or eme-gi7
Dumu-gir15 = Sumerian man, also written dumu-gi7, or dumu-gi7-ra
gir15 is sometimes replaced by the variants gi7-ra or gi-ra.
The word gir15 has nothing to do with gi "reed".
To add on to Arnaud Fournet's answer:
Sumerian seems to have had a phonological process much like modern French, where most consonants aren't pronounced at the end of words. This means that gi "reed" and gi "Sumerian" may have been pronounced the same, based on Akkadian transcriptions in lexical lists.
But this changes when endings are added, like the genitive -ak. To my understanding, gi "reed" had no "hidden" final consonant (auslaut), so its genitive would be gia, while gi "Sumerian" had one, so its genitive would be gira. This is why the word is sometimes transcribed as gi(r). (And the final -k of the ending is then also not pronounced, unless you add another ending after it, etc.)
Auslauts aside, though, according to the PSD the word for "Sumerian" was actually ŋir; the difference between ŋ and g is concealed by the Akkadian transcriptions, since Akkadian lacked
/ŋ/. And as Fournet says, ŋir is written with the sign GIR₁₅ aka GI₇, while gi is written with the sign GI. This doesn't rule out the possibility that ŋir is historically derived from gi, but it implies that the Sumerians didn't think of it that way.