It wasn't exactly one or the other; both the celebrity of his uncle and his promises to guard France against revolutionary "Reds" certainly helped Bonaparte win the presidency in 1848.

Edward Berenson has a great essay about the Second Republic and the Election of 1848 in The French Republic: History, Values, Debates (2011). One of the things that Berenson points out is that at the time, nobody within the assembly - itself predominantly moderate - thought Louis-Napoleon Bonaparte had a chance of winning. Everyone assumed that Louis-Eugène Cavaignac, who was a more republican leaning candidate, would win in a landslide. Bonaparte won over Cavignac by a difference of nearly 5.5 million to 1.4 million votes, with other candidates (Lamartine and Ledru-Rollin were the only other viable ones) not even coming close to those numbers.

Historians have typically explained this upset by pointing out that Bonaparte benefitted from the support of two important voting blocks: monarchists and the peasantry. Many right-wing voters decided to vote for Bonaparte to ensure order in the midst of the early chaos in the Revolution, and the rural peasantry did the same, thinking that Bonaparte would be the best option to instill order and wishing to vote against local notables who supported Cavignac. Karl Marx comments on this in his famous work "The 18th Brumaire and Louis Bonaparte" where he chastizes both the French peasantry for voting against their economic interests and elites for throwing away their liberties to secure their private property. It's a bit of a joke among French historians that the gist of that work is, "The stupid peasants don't know what's good for them!" It also didn't help, though, that throughout the early 19th century in France Bonapartism and Republicanism were often thought of as historically interconnected. For many peasants, the liberals in the assembly didn't represent the tradition of the Republic, or they represented its worst aspects (the Terror). A Bonaparte, however, did, at least in cultural memory.

One aspect that's often overlooked, however, was that Bonaparte also had some support from the industrial working class because he had espoused some socialist economic views in earlier days (he proceeded to implement none of them during both his presidency and the empire). Typically, though, the focus has been on the peasants' support of Bonaparte and their importance in bringing about the Second Empire, even though they would tend to support left-wing candidates in the future. Part of this has been because Marxist historians have paid particularly close attention to the interplay between rural peasantries and urban working classes in history. New scholarship has viewed the Second Republic not so much as a "failure," but as an important historical event that set many democratic and political precedents for the Third Republic.