In the standard Big-Bang nucleosynthesis (BBN) model, the primordial $^7$Li abundance is overestimated by about a factor of 2--3 comparing to the astronomical observations, so called the pending cosmological lithium problem. The $^7$Be($n$,$alpha$)$^4$He reaction, which may affect the $^7$Li abundance, was regarded as the secondary important reaction in destructing the $^7$Be nucleus in BBN. However, the thermonuclear rate of $^7$Be($n$,$alpha$)$^4$He has not been well studied so far. This reaction rate was firstly estimated by Wagoner in 1969, which has been generally adopted in the current BBN simulations and the reaction rate library. This simple estimation involved only a direct-capture reaction mechanism, but the resonant contribution should be also considered according to the later experimental results. In this work, we have revised this rate based on the indirect cross-section data available for the $^4$He($alpha$,$n$)$^7$Be and $^4$He($alpha$,$p$)$^7$Li reactions, with the charge symmetry and detailed-balance principle. Our new result shows that the previous rate (acting as an upper limit) is overestimated by about a factor of ten. The BBN simulation shows that the present rate leads to a 1.2% increase in the final $^7$Li abundance compared to the result using the Wagoner rate, and hence the present rate even worsens the $^7$Li problem. By the present estimation, the role of $^7$Be($n$,$alpha$)$^4$He in destroying $^7$Be is weakened from the secondary importance to the third, and the $^7$Be($d$,$p$)2$^4$He reaction becomes of secondary importance in destructing $^7$Be.
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