I'm actually reading a biography on the Empress Dowager Cixi at the moment[1], and have just got past the Boxer Rebellion, so here's some quite fresh thoughts (I'm assuming you don't want to go so far back as to avoid the Treaty of Shimonoseki which is quite doable if you have Cixi give the Guangxu Emperor a more radical education that makes him more reformist/more likely to continue to listen to her and maintain the reforms and naval purchases of the regency).
1. Avoiding the Boxers altogether: German troops do not go into Shangdong on the flimsy pretence of a murdered missionary who's killers were already being punished and burn lots of local houses during a famine and period of general anti-foreigner tensions following the sudden house arrest of the Emperor at the Summer Palace (this itself was due to the, probably correct, suspicions that he was aware and somewhat approving of the 1898 plot organised by Kang Youwei to assassinate the Empress Dowager. Interestingly enough that would probably have led to the Emperor becoming a Japanese puppet or even Kang deposing him and crowning himself Emperor as he'd voiced aspirations of doing so earlier on, or possibly both). This is, IMO, the weakest option as it really relies on Kaiser Wilhelm II being less stubborn which is very unlikely.
2. Far easier: The Dowager Empress (effectively ruling China at the time) decides to follow both her first instincts and the advice of her historic supporters and continues her policy of stopping the Boxers in Zhili province. By the time she wavered and decided to listen to those advocating that the Boxers be armed they had already been cleared from Shangdong and were being rapidly reduced in Zhili so would have been put down by the 2 month deadline preventing full-on Western intervention.
2b. As a sub-idea of this, the Western powers don't send gunboats to the Dadu forts and are somehow persuaded to give Beijing a chance to prove that they are putting down the Boxers without the need for western coercion. This is a bit trickier, but if done would mean that the Dowager Empress isn't going to be tempted to call the western power's bluff using the Boxers as an army.
As for effects, I think this is going to be a double-edged sword. On the one hand, with Italy the previous year Cixi had signalled that she wasn't going to be bullied into giving in at the first hint of a threat meaning that any further concessions from China would need to be forced by a power absolutely committed to getting them and would require a war in all likelihood- this basically means either the Germans or the Japanese as the other powers didn't really want to partition the country and had too many investments tied up in China to risk bankrupting her (indeed, this is the main reason why China wasn't partitioned after the Boxer Rebellion OTL and why the indemnity was rounded down a bit). On the other hand, the occupation of Tianjing and the experiences of the war made the public a lot more receptive to adopting western practices than they had been, and Cixi's position was actually somewhat strengthened by her conduct in the war, which taken together may mean that the largescale reforms of 1901-1908 face more opposition (which may be cancelled out by having more resources to carry them out), and Cixi herself was emotionally changed quite strongly by the events of 1900.
As such, I think China will probably continue to develop and reform at a somewhat slower pace, but remains generally more stable. Russia and Japan may still go to war, though it would likely mean less Japanese in Manchuria as a result, and I think we may see two general flashpoints. The first is continued German attempts to gain full control of the Shangdong peninsular which may well feed into the causes for *WWI (and perhaps this China, still slowly modernising with the Qing in charge might get more than the Japanese out of the concessions in China at this time). The Second is Japans continued aims to try and create a 'union' with China which would effectively have puppetised the country, and might have led to a general European war with Japan or, more likely, the reduction of China through removal of outlying areas to combat this new power if it had gone through (which is admittedly highly unlikely). A second war with Japan is a distinct possibility, though the Europeans may intervene on China's side at this point, especially if Russia has already been defeated by them.
[1] Jung Chang Empress Dowager Cixi- the Concubine who launched Modern China. It's very revisionist on the traditional view on Cixi as a despot. May be a bit rosy at times, but there's some pretty persuasive evidence in the form of documents not previously available for study. Certainly the concept of the Guangxu Emperor as a great reformer being held back by Cixi falls apart at the seams when faced with the fact that the one period when reforms in China were definitively cancelled and rolled back was when he attained his majority and she retired from politics, and that's the exact period which caused China to fall so drastically far behind Japan in the naval race leading up to the First Sino-Japanese War.