In the following phrase, what does 好 mean in this context?
The correct word boundary should be 好一番, which means a lot of. In your sentence, "we have to go through a lot of struggles."
经历好：V＋好，好 functions as complement of result 结果补语 cf.http://blog.sina.com.cn/s/blog_5f9ec9b00100i4jg.html 3、动词+好:晚饭做好了没有？我要写好毛笔字。这件事情我已经想好了。我们说好了一起回家。
Yufa! A Practical Guide to Mandarin Chinese Grammar， verbs associated with 好 meaning "properly" or "securely" are 学、做、放、带、拿、准备，etc.e.g.各位旅客请在座位上坐好，并且系好安全带，飞机就要起飞了。大家都准备好了吗？可以出发了吧？ When 好 indicates completion of a task,an assignment,piece of work it shares similarities with 完，therefore they sometimes are interchangeable. However 好 tends to mean the completed work can be shown, whereas 完 indicates that once sth. is done it's gone.: verbs associated with this meaning are 作、写、画、洗、翻译，etc.e.g.王老师花了三年，才把那语法书写好。Compare 今天的功课真多，我写了三个多小时，还没写完。
here 挣扎 desperate struggle， 番 classifier for the frequency or number of times an action or deed is carried out - mostly used in idiomatic phrases, 好一番挣扎 numeral measure word combination 数词和量词 e.g. 费了好一番力气 I wasted a lot of effort, therefore 经历好一番挣扎 can mean "lived through/experienced/went through a good deal of desperate struggle"
'好' means 'quite' in ‘好一番’. '好' is adverb here.