Support:  An Advanced Google search in English over the last year for Beersheba excluding Be'er Sheva and Wikipedia gives 59,200 hits (page through to the last result and it turns out to be 557 unduplicated hits).  Restrict that to sites in Israel in English, and you get 2,520 hits (465 unduplicated hits).  A similar search for Be'er Sheva excluding beersheba and Wikipedia gives 26,100 hits (which turns out to be 533 unduplicated hits) or, when restricted to sites in Israel 9,850 hits (which turns out to be 511 unduplicated hits).  This suggests that useage in Israeli English certainly favours Be'er Sheva, but not by as convincing a margin as might be expected (52 per cent to 48 per cent).  In the world as a whole, Beersheba still has the edge (but only by 51 per cent to 49 per cent).  However, while all the results for Be'er Sheva are for the town, there are a significant number of Beersheba hits that relate to other things, including an Australian army reorganisation plan, an aid project in Senegal, a farm in England, a religious campsite in the United States, a self-catering resort in New Zealand, an arts and crafts fair in the United States, a town in the United States, a woman in India, a printing company in the Philippines, an architectural restoration competition in the United States, a man in the United Arab Emirates, a community development programme in Uganda, a school in India, another school in India, a religious medical ministry in the United States, a man in the Philippines and a gardening company in New Zealand (and that's just the first 20 pages).  On balance, it looks like the Hebrew spelling now has the edge.