A suggest you seek previous publications of NAMES: A Journal of Onomastics, published by Taylor & Francis. It is a journal dedicated to the study of names with different theoretical approaches.
Notably, proper names are not simply arbitrary labels but tags carrying significant functions and meanings. Therefore, you have a very interesting topic there. As for the instruments, you can download a slide share (2012) comprising 68 clip slides titled "9 METHODS of DATA COLLECTION". I am sure you find it very useful.
If you want to look at what people do when they choose names, find the best records, f.i. church registers for a given period. If you want to know what they think they do, questionnaires may serve your purpose, but a lot of people won´t fill in or answer questionnaires - at least in my country. So with questionnaires the resulting collection may be skewered.
The instrument to be used will be determined by the nature and focus of the study. For instance, if you are interested in the syntax of personal names, then class register, admission list, graduation lis, church register, etc. will do. I did a study on the structure of personal names of students from two ethnic groups in Ghana. I used the matriculation and graduation brochure of a public university. these contained names of the matriculants and the graduants and that constituted my data. Again, if you are interested in the semantics of personal names, or motivations behind personal names, the instrument and approach will be different. A friend of mine is investigating the morphological processes involved in the names of babies in postmodern Ghana. He visited a creche for the data, but for the processes he had to conduct an interview. He for instance came across a name 'Jismine'. From the parents, the researcher realized that it is from 'Jesus is mine' --J + is + mine.. I hope you find my thoughts useful
In Norway the public population register publishes name statistics every year. Norway has an old register of personal names not used outside the Nordic countries, but these names have been receding in recent years. The current challenge is the heterogenous spelling of names used in the immigrant communities, f.i. a dozen different spellings for "Muhammed".