ryllo -- The recommendations that the other members have made are very good. For long hours on a PJ, stay away from Panasonic - they tend to be anal about running their PJs for more than 6 hours per session (from what I've read).
I have both a Mitsubishi HC3 (960 x 540, LCD) that I bought for $600, four years ago (it's my backup unit). I just bought a Mitsubishi HC5500 (1080p, LCD) to replace my sick 720p DLP PJ. I usually run my PJs for 4 to 12 hours per session (6 to 10 hours, typical) with no problems for either the HC3 or HC5500. My HC5500 shows no SDE (screen door) at any distance beyond 3' from my 106" (16:9) screen.
One thing you should know is that some PJ owners report that their Xbox 360 looks much better on a 720p PJ, whereas the PS3 looks better with a 1080p PJ (but is Ok on a 720p PJ). This may effect your decision (as well as save you some money).
At 10' from a 88" - 90" screen, you will get almost no SDE (or none at all) with just about any 720p projector (depending on how good your eyesight is). Therefore, that might help your decision.
Another factor in your decision (as georgeorwell just said) is that most DLP PJs have sealed optics and no filter. With LCD PJ you have an air filter which should be cleaned every 100 hours of use. Without this extra maintenance, you will have the tendency for getting dust blobs on your LCD panels. A low cost 720p DLP would be great for the kids gaming activities.