* First reported use of drone by Palestinian militants
* Mediation offers mount; Israel, Hamas say fight not over
* No sign ceasefire near after week of violence (Adds Tel Aviv area rocket attack)
By Nidal al-Mughrabi and Jeffrey Heller
GAZA/JERUSALEM, July 14 (Reuters) - Palestinian militants resumed rocket attacks on Tel Aviv on Monday after a 24-hour lull in strikes on the Israeli commercial capital, and Israel kept up its air and naval bombardments of the Gaza Strip despite growing pressure for a ceasefire.
The military said it had shot down a drone from Gaza, the first reported deployment of an unmanned aircraft by Palestinian militants whose rocket attacks have been regularly intercepted.
The use of a drone would mark a step up in the sophistication of the Palestinian arsenal, although it was not immediately clear whether it was armed.
Around half a dozen Israelis have been wounded since the start of the week-old offensive, which Gaza health officials say has killed 169 Palestinians, most of them civilians.
International calls for a ceasefire have grown as the death toll has mounted in the worst flare-up of Israeli-Palestinian violence for almost two years, sparked by the murder of three Israeli teenagers and revenge killing of a Palestinian youth.
Israel has arrested three people, two of them minors, over the Palestinian's murder and officials said on Monday they had confessed to burning him alive.
The European Union said on Monday it was in touch with "all parties in the region" to press for an immediate halt to the hostilities, a day after U.S. Secretary of State John Kerry offered to help secure a Gaza truce.
Egypt and Qatar were also involved but peace efforts were complicated by Hamas's rejection of a mere "calm for calm" in which both sides hold their fire in favour of wider conditions including prisoner release and an end to Israel's Gaza blockade.
The Israeli army said its aircraft and naval gunboats attacked dozens of targets in the Gaza Strip and that Palestinian militants fired more than 20 rockets into Israel, slightly wounding a boy in the town of Ashdod, where a home was damaged. Palestinian health officials said at least 20 people in Gaza were wounded.
But Israel did not carry out a threat to step up attacks against rocket-launching sites it said were hidden among civilian homes in the town of Beit Lahuiya after urging residents there to leave. A U.N. aid agency said around a quarter of the town's 70,000 residents had fled.
Tel Aviv experienced a rare lull in morning rocket strikes, but they resumed during the evening rush hour, with the Iron Dome missile interceptor system going into action. There were no reports of casualties or damage.
Hamas, the Islamist group which runs Gaza, said its armed wing had sent several locally-made drones to carry out "special missions" deep inside Israel.
A military spokesman said the drone was intercepted near the port of Ashdod by a U.S.-built Patriot missile, used largely ineffectively by Israel against Iraqi Scud missiles in the 1991 Gulf War.
The force was trying to locate debris in the area about 25 km (15 miles) north of Gaza, and determine whether it had carried explosives.
"Hamas is trying for an achievement at any price," Israeli Defence Minister Moshe Yaalon, referring to the drone, said in a statement. "We will continue to pummel Hamas and other terrorist organisations until the safety of Israeli citizens is ensured."
An Egyptian-brokered truce doused the last big Gaza flare-up in 2012, and U.N. Secretary General Ban Ki-moon told Egyptian President Abel Fattah al-Sisi in a phone call that his country is the most credible party capable of persuading both sides to stand down, an official Egyptian statement said.
But Cairo's government is at odds with Islamist Hamas, complicating a mediation bid with the group, an offshoot of the now-outlawed Egyptian Muslim Brotherhood.
Asked if Egypt was mediating, Foreign Ministry spokesman Badr Abdelatty said only that Cairo was "in close contact with the Israelis and all Palestinian factions as well as with regional and international countries".
He said he did not want to predict whether those efforts were moving Israel and Hamas close to a ceasefire.
A Hamas politbureau member said Kerry called the foreign minister of Qatar this week, asking him to mediate with the Palestinian movement. A Qatari government source said, however, that Hamas had unrealistic conditions for a ceasefire.
"Qatar is the only one that reached out to us," Hamas official Ezzat al-Rishq said in Doha. "I wouldn't say it's mediation - it's still too early - they have just opened a line of communication with us, but there is no clear plan on what form of mediation this will be."
SEVENTH DAY OF WARFARE
Al-Mezan, a Gaza-based Palestinian human rights group, said 869 Palestinian homes have been destroyed or damaged in Israeli attacks over the past week.
Hamas and Islamic Jihad, the second-most potent Gaza faction, have made clear they would not accept a mere "calm for calm" where both Palestinian fighters and Israeli forces stand down.
Hamas leaders have said a ceasefire must include an end to Israel's Gaza blockade and a recommitment to the 2012 truce agreement.
In addition, Hamas wants Egypt to ease restrictions it imposed at its Rafah crossing with the Gaza Strip since the military toppled Islamist president Mohamed Mursi last July.
Hamas has faced a cash crisis and Gaza's economic hardship has deepened as a result of Egypt's destruction of cross-border smuggling tunnels. Cairo accuses Hamas of aiding anti-government Islamist militants in Egypt's Sinai peninsula, an allegation the Palestinian group denies.
For its part, Hamas leaders said, Israel would have to release hundreds of the group's activists it arrested in the occupied West Bank last month while searching for three Jewish seminary students who it said were kidnapped by Hamas.
The detainees include more than 50 Hamas men who were freed from Israeli jails in 2011 in an exchange of more than 1,000 Palestinian prisoners for Gilad Shalit, an Israeli soldier held by the movement.
Rocket fire from Gaza increased during the West Bank dragnet and tensions were further inflamed when the bodies of the three Israeli teens were discovered on June 30. Two days later, a Palestinian youth was killed in Jerusalem in a suspected Israeli revenge attack.
The current Israeli offensive, which began last Tuesday, has claimed the lives of at least 138 civilians, including 30 children, Gaza health officials said.
There have been no fatalities in Israel since the border hostilities intensified. Iron Dome, which is Israeli-built and partly funded by the United States, has intercepted many of the rocket salvoes.
But the persistent rocket fire has disrupted life in major cities, paralyzed vulnerable southern towns and triggered Israeli mobilisation of troops for a possible Gaza invasion if the Palestinian rockets persisted.
While allowing for an eventual diplomatic solution, an Israeli official said Israel would, for now, pursue its military offensive "to restore quiet over a protracted period by inflicting significant damage to Hamas and the other terrorist groups in the Gaza Strip".
In Ashdod, where Israel's chief peace negotiator, Justice Minister Tzipi Livni was inspecting rocket damage, her spokeswoman refused to allow any questions about a truce. She said anything Livni might say about a potential ceasefire could derail any diplomatic efforts to achieve one.
(Additional reporting by Maayan Lubell and Dan Williams in Jerusalem, Michael Georgy in Cairo and Amena Bakr in Doha; Writing by Jeffrey Heller)