Yes, there is a definite benefit. All SAs (other than USCGA, which doesn't require noms) must "slot" or assign appointees to a source of nominations. Thus, bluntly stated, the more noms you have, the more places the SA can slot you and thus the greater your chances of receiving an appointment.
To explain, take two candidates applying to USNA:
Candidate A has only a nom from her Senator. Thus, USNA can only slot that person to the Senator (she counts against the Senator's total number of mids that can be at USNA at one time). If USNA finds someone more competitve than Candidate A on the Senator's slate or if the Senator has named someone else as his principal nominee, she then goes into the National Pool to compete and, if offered an appointment, will likely be slotted to SecNav.
Candidate B has a Pres nom, a nom from his Rep and a nom from his Senator. USNA now has 3 places where they can slot him before sending him to the National Pool. That gives USNA more options and gives Candidate B a better chance of securing an appointment before heading to the Pool.
Also, MOC noms are competitive (as opposed to Pres. which are not). Securing several noms can demonstrate your competitiveness to the SA.
Note that some MOCs -- primarily those in very competitive regions of the country -- will not nominate an individual who already has secured a nom. They basically get together and compare notes so that as many constituents as possible receive noms. The SAs are aware this happens, so it's not a negative if you only receive a single nom.
Finally, yes, all it takes is one nom.
As an aside, if you receive an LOA, there is no benefit in securing more than one nom b/c, if you have the LOA and are fully qualilfied and have one nom from any source, you will be offered an appointment.