No, mobile phones do not attract lightning strikes.
NPR specifically addressed this BMJ article in Debunking the Cell Phone-Lightning Connection on August 2, 2006:
You may have heard something a few weeks ago about how getting struck by lightning is even worse if you're carrying a cell phone. Many news organizations reported about the cell phone-lightning connection that appeared in the British Medical Journal.
But, did you hear anything about the follow-up letters in the same journal, the ones that debunked the first report? Probably not.
The letters referenced in this piece appear in BMJ: British Medical Journal, Vol. 333, No. 7558 (8 July 2006), p. 96. In addition to the basic physics of how electricity works, these letters also cite the low quantity of metal in mobile phones, the lack of a statistical rise in lightning strikes as mobile phone usage has become ubiquitous, and that the original article's claim that the Australian Lightning Protection Standard advises against mobile phone usage during thunderstorms is incorrect - it advises against corded phone usage during thunderstorms and suggests mobile phones as a recommended alternative. In the US, the NOAA makes this same recommendation (emphasis mine):
However, when inside during a thunderstorm, avoid contact with anything that could conduct a lightning strike to you, including anything that plugs into a wall outlet, corded phones, plumbing, metal doors, and window frames.
This means do not take a shower or bath during a thunder storm. Battery-operated computers and cellphones are fine.
I was also able to find a more extensive 2014 paper titled "On the Absurdity of 'Mobile Phone Triggered Lightning'", by Li Daxiao of the Meteorological Bureau of Santai County, China, and published in Meteorological & Environmental Research; Nov2014, Vol. 5 Issue 11, p42-45. (ISSN 21523940; I was not able to find a DOI or an open-access link to the paper.)
The abstract of the paper reads:
That making phone calls had nothing to do with lightning strike was proved from the aspects of radio wave characteristics, receiving and transmitting frequency, power as well as the electromagnetic induction of mobile phones. And through assuming mobile phone could trigger lightning, the probable death toll disproved the absurdity of mobile phone triggered lightning. And we concluded that making phone calls in thundery days would not increase the probability of lightning stroke, and calls could be made in thundery days as long as in the safe position.
After examining the question from seven different angles, the "Conclusion" paragraph states:
The absurdity of "mobile phone could trigger lightning stroke" not only lies in it ignores the nature of the events but only focuses on the surface, but more importantly it violates the scientific law and is entirely a misunderstanding of the knowledge of lightning protection and physics. In no case could mobile phone trigger lightning stroke. People got stricken by lightning when making phone calls is simply because they stand at the wrong place in the improper time. As long as in the safe place, people can make phone calls even in thundery days.
