For all round general purpose reasonably "future proofed" computing needs I tend to recommend a minimum Core I5 (if you can afford it). That's what I put in the PC World laptop selection criteria and the HP was the only one within the budget. Pentiums are a bit bargain basement, but as you say are usable for very light use. I bought a HP Stream Pentium for my wife nearly 3 years ago and she's quite happy with it, although it does slow down dramatically when she overindulges with upteen youtube tabs open in chrome.
Speaking of chrome, Chromebooks would also fit comfortably in the budget and although I'm not a fan of them myself there are quite a few folks in here who like them and it might be a suitable alternative.