We plan to be in Jerusalem Feb 6-9 on our own. We were on a tour for 3 days earlier in the week in Jerusalem. Are there any suggestions about getting to Bethlehem with the political issues at hand? What other places should we see while there?
Don't think I can help with a cheap place since we stayed at the Sheraton (on points) when we were there in June 2008. However, getting to Bethlehem was very easy. Exit the old town through the Damascus gate, cross the street to where the buses leave from and ask for the right bus. The bus drops you at the security wall - cross it after showing your passport, pick up a cab on the other side and it is a 5 min ride to the center of Bethlehem.
On the return we took a bus that dropped us directly in the center of Jerusalem (we used lonely planet as a guide book which had all the details).
We traveled independently with our kids throughout that area of the Middle East including Lebanon and Syria and didn't encounter any security issues (note - this was less than a month after some of the disturbances in Beirut in May 08). I would think you will have no issues since you are just in Israel.
I don't know what you consider "budget", but a number of years ago we stayed at the Jerusalem Hotel in East Jerusalem, near the Damascus Gate. It wasn't expensive then (otherwise we wouldn't have stayed there) and having just had a look at their website it seems moderately priced to me. It is a gorgeous, small family-run hotel, and our room was large & well appointed. Breakfast was in the walled garden. The whole experience was exotic and "middle-eastish". People we met who were staying at a large, modern U.S. style hotel in West Jerusalem moved to our hotel as soon as they saw it. www.jrshotel.com