If you use MyISAM, you can use full-text indexing. See this tutorial
If you use a different storage engine, you can use a third-party full-text engine such as sphinx, which can act as a storage engine for mysql or a separate server that can be requested.
With full MySQL text indexing, searching in AJ Kelly will match AJ Kelly (don't confuse the questions, but A, J and AJ will be ignored as they are too short by default and this will match Kelly.) Usually Fulltext is much more forgiving (and usually faster than LIKE '% string%'), because it allows partial matches, which can then be ranked by relevance.
You can also use SOUNDEX to make the search more forgiving by indexing the phonetic equivalents of words and looking at them, applying SOUNDEX to your search terms and then using them to find the index. With soundex, mary , marie and marry will match, for example.
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