I have heard that Muslims believe that God does not have a name. Apparently, when they refer to Him as Allah, this is just a common noun that means "God" in Arabic (and has similar cognates in other Semitic languages), but is not a personal name. Is this claim true? What are the implications of worshipping a God that does not have a name?
In fact, this claim is absolutely wrong. Allah has lots of names that have been stated in Quran and other supplication from Prophet Muhammad (pbuh) and twelve Imams.
For example, at least 99 names have been mentioned in holy Quran. Or, 1000 names exist in Jowshan Kabir (جوشن کبیر) supplication. And, Quran explicitly says:
وَلِلَّهِ الْأَسْمَاءُ الْحُسْنَىٰ فَادْعُوهُ بِهَا ۖ وَذَرُوا الَّذِينَ يُلْحِدُونَ فِي أَسْمَائِهِ ۚ سَيُجْزَوْنَ مَا كَانُوا يَعْمَلُونَ - 7:180
And to Allah belong the best names, so invoke Him by them. And leave [the company of] those who practice deviation concerning His names. They will be recompensed for what they have been doing.
I am not aware of any belief that "God does not have a name". And on the face of it, this seems contradictory to several statements from the Quran like 1:1, 2:114, 5:4, 6:118, 6:119, 6:121, 7:180, 11:41, 17:110, 20:8, 22:28, 22:34, 22:36, 22:40, 24:36, 27:30, 55:78, 56:74, 56:96, 59:24, 69:52, 73:8, 76:25, 87:1, 87:15, 96:1.
Allah has several names. Some of these, including Allah, are personal names, in that they are specific to Him and not given to anyone except the real God.