Can a person travel to Israel with a European passport but an Arabic name?

How much trouble they would expect to have would depend on their individual circumstances.

Have they visited many Arab countries and do they have stamps from these evident in their passport?

If they simply have an Arab name and have a European passport with nothing to suspect anything other than an innocent tourist visit, they should expect no problems - except for a grilling at the airport. Your friend should expect them to have a bit of an attitude, they normally do. But as long as she has proof that she is travelling as a tourist - perhaps hotel booking or other evidence of a holiday, she should be ok. It may be a bit of a delay at the airport while they check on her, that's all.

When my husband and I visited Israel, we were British born and bred and they even had an attitude with us, tipping the contents of all our bags unceremoniously onto a desk and roughly going through them and leaving us to take ages to try and get it all back again. Sometimes it seems that they just enjoy having power over you. But you never mess with them. Just don't expect a friendly reception!

Oh, and if she has been to any Arab countries and it shows in her passport, it might be worth her trying to get a second passport. We knew people who did that because their job required frequent trips to Israel and Arab countries.

Your name does not matter, if the passport is valid and you have all the requirements such as visa in case your country needs one, you are OK to go. Even inside Israel you wont have any problems, I can not say the same think for an Israeli name traveling to an arab country but thats another story.

Names shouldn't make any difference. As long as they've got the passport they should be fine.