HauteFinance

As far as I know, there is no dissertation as part of the MSc EME (both the 1-year and 2-year programs, for that matter). Written coursework during the term is likely to be formative; most modules are graded solely on an exam, so the work probably won't be "official" enough to count for what you need. I'm also skeptical that the department would be gung-ho on a voluntary dissertation, but you would have to contact them for a definitive answer. The MSc Economics does include a dissertation, so that might be a better fit for the requirements.

The MSc Economics is probably the most well-rounded of all the options; it's still rigorous and gives you a lot of flexibility in terms of post-study options. For Oxford, I don't know much about the school or the program, so I can't really make a judgement. It has a great reputation and a good mix of courses, but finding their placement statistics would be more helpful than my guess. How the research track works is that you apply for the regular program, and then could be accepted after pre-sessional exams based on results/background. Even then it's not a guarantee for PhD Finance admissions, but completing the research track would allow you to finish the MRes in one year instead of two. I just called it brutal because the indicative reading list for the financial econometrics course look like it assumed quite a bit of prior knowledge. Career-wise, I don't think you can go wrong with either Oxford or the LSE. In regards to general preparation and the real analysis requirement, there are other folks on the forum much more qualified to give that advice. I'm in neither the economics nor finance department at the LSE, so I'm only useful for the general stuff.

Depending on the structure of the master's program, you may not even have any grades/completed work for them to base their letters on at that point. From personal experience, 1-year programs really aren't ideal if you are both looking to apply for PhD right away and need recommendations. If you have the chance for an RA opportunity, it could be a really beneficial option.

Based on your interests/goals, I'd say to consider dropping "Accounting and Finance", "Finance and Private Equity(?)", and "Finance" from the list. They're all far from bad programs, but you won't be able to take any rigorous econometric/economics courses, which would be a benefit going forward if you're hoping to have the option to go for a PhD. From what I can tell, those three programs tend to be more on the side of churning out graduates for the consulting/investment banking industry [The LSE does have a bit of a reputation for that]. EME would be the only one where could get an Analysis course, since you can take some modules from the Mathematics department with approval. That aside, "Finance and Economics" matches pretty well with what you want to achieve. You'd have access to professors in both departments, and could apply for the Research track if you're really looking for more advanced micro/independent work. Fair warning though, that track is basically the equivalent of the 1st year of their PhD program so it will be brutal. That's also one of the more competitive programs though (both tracks), but you'd have a pretty decent chance with the amount of mathematics and research experience you have. In regards to careers, what country are you looking to work in afterwards? If you're planning to go back to the US, the brand name is probably worth more than the specific department to the average employer.

The best answer would really depend on what specific kind of work you're interested in pursuing. In regards to program at the LSE, it's not that it isn't rigorous so much as it is a completely different field. It is taught by the Sociology Department and there are no options to take courses from outside that department. It simply won't give you the postgraduate economics training that economic consulting firms/government organizations/investment banks are looking for in applicants. If you just want any job in London (that is, non-econ focused), the LSE does offer a brand name advantage. However, keep in mind that you'll be competing with people who completed MSc in finance or management, which may be more attractive to the average firm. To give you an idea, the top three destination areas for the Sociology Department in 2013-2014 were "Research and Education", "Charities, Development and International Organizations", and "Media and Publishing". If you do go this route, you'll need to start searching for jobs as soon as you arrive in the UK. From personal experience, it seems like most of the large recruitment drives occur around November. I know absolutely nothing about the program at Bocconi, but if you want a job in economics, a degree in sociology (while most likely interesting) will not make your search any easier.

Currently enrolled in the MSc Economic History at the LSE, so I figured I'd chime in. As far as I know it's the only university with a stand-alone economic history department, and now offers postgraduate options in both "Economic History" and "Quantitative Economic History". The former is fairly flexible in terms of the course options you have, while the latter is built around the standard postgraduate economics core. Your friend could study both topics within the department (there's a pretty wide range of faculty, honestly), but, as chateauheart and tm_member pointed out, the necessary skill sets don't overlap all that well. Oxbridge has some programs as well, but I believe they are in "Social and Economic History" so you're looking at a slightly different environment. For the US, things are a bit trickier. Before applying, they'll really need to decide if their primary methodology will be economics or history, as both departments often have people doing related research. In terms of economics departments (adding to the earlier list), I know Yale does quite a bit on EH and I think Vanderbilt has it as a field option. I don't know anything about the program at Maryland, but they seem to offer a "Trade, Institutions, History, and Political Economy" field. In terms of history departments, there will be a scattering of folks at a good number of universities, but where they apply would depend much more heavily on the specific type/country/era of economic history in which your friend is interested. Temin has a working paper from 2013 on "The Rise and Fall of Economic History at MIT" that gives a good feel for the situation in the US. Overall rankings can be a bit difficult to make because, depending on how far they shift to the historical side, that can almost involve a completely different job market.