I think there are 2 non-Hamiltonian graphs with 5 vertices and exactly 7 edges. There are a total of 4 graphs with 5 vertices and 7 edges. If I run (in version 9):
n = 5; Count[Table[HamiltonianGraphQ[ RandomGraph[{n, Binomial[n - 1, 2] + 1}]], {1000000}], False] Mathematica returns about 250000. I was expecting about half a million.
Indeed there are four graphs with 5 vertices and 7 edges, two of which have a Hamiltonian cycle:
TableForm[{#, HamiltonianGraphQ@#} & /@ DeleteDuplicates[ RandomGraph[{n, Binomial[n - 1, 2] + 1}, 1000], IsomorphicGraphQ ], TableHeadings -> {None, {"Graph", "HamiltonianGraphQ"}} ] 
However, RandomGraph doesn't sample from that space, but from a larger space that includes many graphs that are isomorphic to each other. If you examine some of the graphs they look identical but under the hood they are different:
Draw 10,000 random graphs and tally taking isomorphism into account and you get this:
Tally[RandomGraph[{n, Binomial[n - 1, 2] + 1}, 10000], IsomorphicGraphQ] 
As you can see the graphs with a Hamiltonian cycle are not available in this set in the same amount as the graphs without such a cycle, hence your results.
In fact, there are a total of 120 different graphs with the 5,7 property:
RandomGraph[{n, Binomial[n - 1, 2] + 1}, 100000] // Union // Length 120
with isomorphicity distributed as follows:
Tally[RandomGraph[{n, Binomial[n - 1, 2] + 1}, 100000] // Union, IsomorphicGraphQ] 
The non-Hamiltonions make up precisely 1/4 of the set. This is consistent with your results.
